Analysis n
831
Μιχάλης Παπαδημητράκης Ανάλυση Πραγματικές Συναρτήσεις μιας Μεταβλητής Τμήμα Μαθηματικών Πανεπιστήμιο Κρήτης
Transcript of Analysis n
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- . (-) , Supremum , , . , - , . - . .
, , Supremum, . , - , Bolzano - Weierstrass , , - .
: - . ( ) , , , , , .
.1. . , n n. , .2. , , 10 - . , , .3. Riemann , , Darboux - . , Riemann . Riemann -, Darboux Darboux Riemann.4. . . : R ( ). Rd - . , , () ., .5. , -
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, . , ( , Riemann , - , ) .6. Peano. , , . Foundations of Analysis E. Landau . (-) () Dedekind., , Cauchy -, .7. . , Cauchy - .8. , , , -. : - , . , , .9. , , . , - (, ). , IQ. . , .
, , :
Mathematical Analysis, T. Apostol.Differential and Integral Calculus, R. Courant.The Theory of Functions of Real Variables, L. Graves.Foundations of Analysis, E. Landau.Principles of Mathematical Analysis, W. Rudin.The Theory of Functions, E. C. Titchmarsh.
. , Graves , .
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2015.
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1 . 11.1 R R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Supremum infimum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Supremum. . . . . . . . . . . . . . . . . . . . . 101.4 , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 . 212.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 . . . . . . . . . . . . . . . . . . . . . . . 312.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.7 . . . . . . . . . . . . . . . . . . . . 64
3 . 703.1 , . . . . . . . . . . . . . . . . . . 703.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.3 . . . . . . . . . . . . . . . . . . . . . . 833.4 . . . . . . . . . . . . . . . . . . . . . . . . . 1013.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.6 Cauchy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4 . 1084.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.3 . . . . . . . . . . . . . . . . . . . . . . . . 1224.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.5 . . . . . . . . . . . . . . . . . . . . . . . . 1344.6 . . . . . . . . . . . . . . . . . . . . . . . . . 142
5 . 1475.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.5 . . . . . . . . . . . . . . . . . . . . . 1785.6 . . . . . . . . . . . . . . . . . . . . . . . 1935.7 Taylor, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2005.8 , . . . . . . . . . . . . . . . . . . . . . . . . 202
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6 Riemann. 2096.1 Darboux. . . . . . . . . . . . . . . . . . . . . . . . 2096.2 . Darboux. . . . . . . . . . . . . . . . . . . . . . . 2136.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2216.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2226.5 . Riemann. . . . . . . . . . . . . . . . . . . . . . . 238
7 . 2467.1 , . . . . . . . . . . . . . . . . . . . . . . 2467.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2527.3 . . . . . . . . . . . . . . . . . . . . . . . 2607.4 Taylor, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
8 . 2838.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2838.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . 2898.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3028.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 3128.5 Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3178.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
9 . 3249.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3249.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3269.3 Weierstrass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
10 . 34210.1 . . . . . . . . . . . . . . . . . . . . . . 34210.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35110.3 Taylor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36510.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 371
11 . 37811.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37811.2 , , . . . . . . . . . . . . . . . . . . . . . 38311.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 39411.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40011.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40411.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40811.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
12 . 43112.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43112.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43812.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44212.4 . . . . . . . . . . 44812.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
13 . 46313.1 Peano. . . . . . . . . . . . . . . . . . . . . . . 46313.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46913.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47513.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
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13.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
14 . 491
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1
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1.1 R R.
R. , - , ,N, Z Q. : N = f1; 2; 3; : : : g. , 0.
- R .1
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1.1. n 2 N, n 2.[] x; y
yn xn = (y x)(yn1 + yn2x+ + yxn2 + xn1): (1.1)
[] nxn1(y x) yn xn nyn1(y x) 0 x y: (1.2)
. [] .[] 0 x y , (1.1) x y, ( ) n yn1 , y x, ( ) n xn1.
BERNOULLI. n 2 N.
(a+ 1)n na+ 1 a 1:
. 2 a 0, y = a + 1 x = 1 (1.2) , 1 a 0, y = 1 x = a+ 1 (1.2). NEWTON. n 2 N. x; y
(x+ y)n =Pn
k=0
nk
xkynk =
Pnk=0
n!k!(nk)!x
kynk:1 11 .2 Bernoulli 1.1.5 5.4.6.
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(x+ y)n = (x+ y) (x+ y) (n ) , n x y . , k 2 Z 0 k n xkynk x k y n k . , k 2 Z 0 k n, xkynk n k k x. , k n
nk
= n!k!(nk)! =
n(n1)(nk+1)k! :
k 2 Z 0 k n nk xkynk., , jxj x -
x 0. , , jx yj 4 x y. jx yj x y.
, .
1.2. [] a > 0, :
jxj a a x a:jxj < a a < x < a:
[] x; y :
jx yj jxj+ jyj:. x y 0 0. , .
. R,
R = R [ f1;+1g: , +1 , 1 1 +1. :
1 < x; x < +1; 1 < +1:
(+1) = 1 ; (1) = +1 :
(+1) + x = +1; x+ (+1) = +1; (+1) + (+1) = +1;3 Newton 1.1.5 5.4.8. -
, .4 .
, , d- . , 9 10. 11 .
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(1) + x = 1; x+ (1) = 1; (1) + (1) = 1:,
(+1) + (1); (1) + (+1)
.
(+1) x = +1; x (1) = +1; (+1) (1) = +1;
(1) x = 1; x (+1) = 1; (1) (+1) = 1:
(+1) (+1); (1) (1)
.
(1)x = 1; x(1) = 1 x > 0;
(1)x = 1; x(1) = 1 x < 0;(1)(1) = +1; (1)(1) = 1:
(1)0; 0(1)
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1+1 = 0;
11 = 0:
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1x = 1 x > 0; 1x = 1 x < 0; x1 = 0:
x0 ;
10 ;
11 ;
11
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j+1j = +1; j 1j = +1:5 1.8,
2.3.28 2.12.
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R R (a;+1), [a;+1), (1; b), (1; b] (1;+1) fx jx > ag, fx jx ag, fx jx < bg, fx jx bg R, -. R , , (a;+1], [a;+1], [1; b), [1; b],[1;+1), (1;+1], [1;+1].
R R . - ( ) ( ) . -, (+1)+ (+1) = +1. , , - . (+1) (+1) . , ( ) , , . (+1)0 , , .
a A , , . , , -, , a 2 (3;+1] a +1. A [1; 2] A 1.
.
1.1.1. 6 a x b a y b, jxyj ba .
1.1.2. x y < 0 z w < 0, 0 < yw xz. : a; b 0, ab 0.1.1.3. [] b1; : : : ; bn > 0. l a1b1 ; : : : ; anbn u, l a1++anb1++bn u.[]7 1; : : : ; n 2 N. l y1; : : : ; yn u, l 1y1++nyn1++n u., w1; : : : ; wn > 0 w1 + + wn = 1. l y1; : : : ; yn u, l w1y1 + + wnyn u.1.1.4. - x : jx+1j > 2, jx1j < jx+1j, xx+2 > x+33x+1 ;(x 2)2 4, jx2 7xj > x2 7x, (x1)(x+4)(x7)(x+5) > 0, (x1)(x3)(x2)2 0. x x : (1; 3], (2;+1), (3; 7),(1;2) [ (1; 4) [ (7;+1), [2; 4] [ [6;+1), [1; 4) [ (4; 8], (1;2] [ [1; 4) [ [7;+1).1.1.5. Bernoulli Newton .
6 , .7 y1; : : : ; yn, yk k , -
, 1y1++nyn
1++n =1
1++n y1 + +n
1++n yn = w1y1 + + wnyn;
wk = k1++n yk, yk y1; : : : ; yn. , w1; : : : ; wn > 0 w1 + + wn = 1, w1; : : : ; wn w1y1 + + wnyn y1; : : : yn .
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1.2 Supremum infimum.
. - A. A u x u x 2 A. u A. A l l x x 2 A. l A., A , - l u l x u x 2 A.
: u A, u0 u A , l A, l0 l A. 1.3. [] l x x > a. l a.[] u x x < b. u b.. [] ( ) a < l. x =a+l2 a < x < l. , , x > a l x. , , .[] , u < b. x = u+b2 u < x < b., , x < b u x. . 1.2.1. l a [a; b], (a; b], [a; b), (a; b), (a;+1), [a;+1). l . [a; b], [a; b), [a;+1). l , l a, a -. l a . (a; b], (a; b), (a;+1). l . a -, l a. . , , l (a;+1). , , l x x > a. , , 1.3 l a. (a; b], (a; b), (a;+1) l a ., , [a; b], (a; b], [a; b), (a; b), (a;+1), [a;+1) , a, l a . (1; a]. 1.2.2. , u b [a; b], [a; b), (a; b], (a; b), (1; b), (1; b]. , , - . [a; b], (a; b], (1; b] 1.3 [a; b), (a; b), (1; b). , , , b, u b . [b;+1). 1.2.3. (a;+1), [a;+1), (1;+1) (1; b), (1; b], (1;+1) . 1.2.4. N , 1 , N l 1 . , 1 N N (1; 1]. N .
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SUPREMUM. -, .
supremum R. , , .
supremum. R R N. , R N supremum, 12.
, supremum, infimum. : 0 , , .
INFIMUM. -, .
. -, A.
A = fx jx 2 Ag: A A 0. , . A -, A -., l A, l A, A ., , A - , supremum u0 A. u0 A, u0 A. l A u0, l A u0. u0 A. l A u0. u0 A.. - A infimum A. - A supremum A. infimum supremum A , ,
infA g.l.b.A supA l.u.b.A:
1.2.5. 1.2.1 1.2.2, [a; b],(a; b), (a; b], [a; b) (1; b], (1; b) supremum, b, [a; b],(a; b), (a; b], [a; b), (a;+1), [a;+1) infimum, a.. , , A minimum A. , , A maximum A. minimum maximum A , ,
minA maxA:
[a; b] [a; b) -, - maximum maximum., -, supremum.
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1.2.6. A maximum, supA = maxA, supremum A maximum A., A maxA, maxA A. , maxA A A maxA. maxA A. , A minimum, infA = minA, infimum A minimum A. A = f0g [ [2; 3] [ f4g minA = 0 maxA = 4. infA = 0 supA = 4. 1.2.7. minN = 1, infN = 1. , N : n 2 N N n+ 1 2 N n., N N supremum. N . , , N - . .
1.2.8. A = f 1n jn 2 Ng = f1; 12 ; 13 ; 14 ; : : : g maxA = 1, supA = 1., A : 1n 2 A A 1
n+1 2 A 1n . , A , , 0, A infimum. infA .
. - A , infA = 1. - A , supA = +1.
(: ) . A , . , +1 , , , , , A.
- A supremum infimum. A -, supremum , , supremum +1., A , infimum , -, infimum 1. , - A
infA supA:
, x 2 A infA x x supA , , infA A supA A.
, supA, , +1. , infA 1.
supA infA.
supA (i) (ii):(i) A supA. supA = +1 , supA , supA A. (i) , , :
x 2 A x supA:
(ii) supA A., supA = +1 u, . u A ( A ), x 2 A u < x. +1 A.
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supA > 0, . supA A ( supA A), x 2 A supA < x , ( (i)),
supA < x supA:
x A, supA - . supA A. (ii) , , :
u < supA x 2 A u < x supA:
, infA (i) (ii):(i) A infA.
x 2 A infA x:
(ii) infA A.
l > infA x 2 A infA x < l:
supremum infimum .
A . A -: x1; x2 2 A x1 < x2 x x1 < x < x2 x 2 A. : . 1.4 , - R, .
1.4. - A : x1; x2 2 A x1 < x2 x x1 < x < x2 x 2 A. A ..
u = supA; l = infA;
1 l u +1. , , A [l; u]. x 2 (l; u). x A, x1; x2 2 A x1 < x < x2. , x 2 A. , (l; u) A. (l; u) A [l; u] :
A = (l; u) A = [l; u] A = (l; u] A = [l; u):
A , , infA supA.
.
1.2.1. maxfx; yg = x+y+jxyj2 minfx; yg = x+yjxyj2 .1.2.2. (a;+1) [a; b) .
1.2.3. l a+ > 0, l a. ja bj > 0, a = b.1.2.4. a 1 0 < < 1, a 0.1.2.5. infA = infB supA = supB. A = B;
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1.2.6. - A. [infA; supA] R A.
1.2.7. [a; b];
1.2.8. - A. ( ) supA A, : supA = +1 supA < +1. A infA.
1.2.9. - A. supA 2 A A . infA A.
1.2.10. A = [0; 2],A = [0; 2),A = [0; 1][f2g, ,, -, A u = supA : A \ (u ; u] 6= ; > 0; A \ (u ; u) 6= ; > 0; , , u /2 A; l = infA.
1.2.11. - A u. supA u x u x 2 A. u supA < u x 2 A < x. infA l.
1.2.12. - A;B.[] supA infB x y x 2 A, y 2 B.[] , . A = (1; 0]; B = [0;+1) x y x 2 A, y 2 B. , x y x 2 A, y 2 B. A = (1; 0], B = (0;+1), A = (4;2), B = (2;+1) A = (1; 0), B = [1; 13]., , x y x 2 A, y 2 B. ( ) supA, infB x y x 2 A,y 2 B.[] x y x 2 A, y 2 B > 0 x 2 A,y 2 B y x . supA = infB x y x 2 A, y 2 B. ;[] 0 < x y x 2 A, y 2 B > 0 x 2 A, y 2 B yx 1 + . supA = infB x y x 2 A, y 2 B.1.2.13. - A;B A [ B = R x < y x 2 A,y 2 B. A;B A B. A = (1; ); B = [;+1) A = (1; ]; B = (;+1).1.2.14. - A;B. A supA supB x 2 A < x y 2 B y > .1.2.15. - A, B A B. infB infA supA supB.1.2.16. [] - A, B. sup(A [B) = maxfsupA; supBg inf(A [B) = minfinfA; infBg., , A \ B , sup(A \ B) minfsupA; supBg
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inf(A \B) maxfinfA; infBg. sup(A \B) = minfsupA; supBg inf(A \B) = maxfinfA; infBg;[] - A A = fx jx 2 Ag. sup(A) = infA inf(A) = supA.[] - A;B A+B = fx+ y jx 2 A; y 2 Bg. A+B A = [3; 5], B = [1; 7] A = (3; 5), B = (1; 7) ; inf(A+B) = infA+ infB sup(A+B) = supA+ supB.[] - A;B A B = fxy jx 2 A; y 2 Bg. A B A = [3; 5],B = [1; 7], A = (3; 5),B = (1; 7) A = (1; 5),B = (2; 7) ; A;B (0;+1), inf(A B) = infA infB sup(A B) = supA supB.1.2.17. ;; , inf ;, sup ;; inf ; sup ;;1.2.18. 8 [] f : [a; b] ! R [a; b]. f(a) > a f(b) < b, 2 (a; b) f() = .[] I f : I ! R x 2 I > 0 f(x0) f(x) f(x00) x0; x00 2 (x ; x + ) \ I x0 x x00. f I .
1.3 Supremum.
1.2.4 1.2.7.
1.1. N , , supN = +1.. ( ) N , supN . supN 1 N, n 2 N supN 1 < n. supN < n+ 1 , n+ 1 2 N.
1.1 :
. l > 0, n 2 N 0 < 1n < l.
, l > 0, 1l , N , n 2 N 1l < n , , 0 < 1n < l. 1.3.1. 1.2.8. A = f 1n jn 2 Ng = f1; 12 ; 13 ; 14 ; : : : g infA = 0., 0 A - l > 0 A. , A 0.
1.5 .
1.5. x k 2 Z k x < k + 1.8 2.4.16.
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. x. 1.1 n 2 N n > x m 2 N m > x. l = m, l; n
l < x < n:
:
k 2 Z, k x, k + 1 x. k x k = l, k x k 2 Z k l. , , k x k = n., , k 2 Z k x k + 1 > x, k x < k + 1. k k x < k + 1 . k x < k+1 k0 x < k0+1 k; k0 2 Z. k < k0+1 k0 < k+1, 1 < k0 k < 1. k0 k 2 Z, k0 k = 0, k0 = k.. k 2 Z k x < k+1, 1.5, x [x].,
[x] 2 Z [x] x < [x] + 1: 1.3.2. [3] = 3, [3] = 3, [3:5] = 3, [3:5] = 4.
, , - . , , ( R) Q (R). .YKNOTHTA . a; b a < b r a < r < b.. b a > 0, , , n 2 N 1n < b a. , na+ 1 < nb,
na < [na] + 1 na+ 1 < nb: r = [na]+1n a < r < b.
.
1.3.1. 1.2.3. l a+ 1n n 2 N, l a. ja bj 1n n 2 N, a = b.1.3.2. infimum supremum f(1)nn jn 2 Ng, N [ f 1n jn 2 Ng,f(1)n 1n jn 2 Ng, f 12n jn 2 Ng [ f1 12n jn 2 Ng,
S+1n=1[2n 1; 2n],
S+1n=1[
12n ;
12n1 ].
1.3.3. infimum supremum (a; b) \Q = fr 2 Q j a < r < bg.1.3.4. 1.2.12. A = f 1n jn 2 Ng, B = f 1n jn 2 Ng x y x 2 A, y 2 B. x y x 2 A, y 2 B. A = fr 2 Q j r < 0g, B = fr 2 Q j r > 0g.1.3.5. r a r 2 Q r > b, b a. fr 2 Q j r < ag = fr 2 Q j r < bg, a = b. fr 2 Q j r < ag \ fr 2 Q j r > bg = ;, a b. fr 2 Q j r ag [ fr 2 Q j r bg = Q, b a.
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1.4 , , .
. n 2 Z, n 1 (, n 2 N), an :
an = a a (n ): n 2 Z, n 1 a 6= 0, :
an = 1aa (n ) :
, n = 0 a 6= 0, :a0 = 1:
1.4.3 . . .
.
, - , , . supremum. 1.2. n 2 N, n 2. y 0 x 0 xn = y.. 9 n 2 N, n 2 y 0.
X = fx jx 0; xn yg: , 0 2 X , X -. , y + 1 1 (y+1)n y+1 > y. x 2 X xn y < (y+1)n , ,x < y + 1. X y + 1. X - , supX .
= supX:
, 0, 0 2 X . n = y. n < y. y
n
n(+1)n1 > 0 > 0 1 yn
n(+1)n1 . (1.2)
( + )n n n( + )n1( + ) = n( + )n1 n( + 1)n1 y n: ( + )n y, + 2 X . , X . n y. n > y.
nynn1 > 0 > 0
nynn1 . , (1.2),
n ( )n nn1 ( ) = nn1 n y:
y ( )n. x 2 X xn y ( )n , , x . X . , X . n y. n y n y n = y., , - xn = y y 0. - xn = y. : 1; 2 0, 1n = y 2n = y, 1n = 2n; 1 = 2.
9, 4.4.20 4 n- npy - y. 2.4.12.
12
-
. n 2 N, n 2. y 0, - xn = y, 1.2, n- y
npy:
n = 2, 2py py. npy - y. ,
n- ( - xn = y) np0 = 0
npy > 0 y > 0., - xn = y
y 0, . n , xn = y (i) , npy npy, y > 0, (ii) , 0, y = 0, (iii) , y < 0. n , xn = y (i) , npy, y > 0, (ii) , 0, y = 0, (iii) , npy, y < 0. .
. npy. , supremum. . 1.4.4 , , . .
1.2 . , n 2 N, xn (, , --) [0;+1) [0;+1). . 1.2 xn [0;+1) [0;+1). , y = xn , [0;+1), , [0;+1)., , x = npy [0;+1) [0;+1). , y = xn ( [0;+1)) x = npy . , y = xn , x = npy .
- - . .
1.6. n; k. npk k n-
. , npk , .
. k n- , m 2 N k = mn. n
pk = m , , .
, npk , n
pk = ml , m; l 2 N. -
, , m; l > 1. n
pk = ml l
nk = mn. , > 1 , . , l lnk, mn = mn1m. l - > 1 m, l mn1 = mn2m. , l mn2 = mn3m. , l m0 = 1. l = 1, k = mn k n- .
, , R .
1.7. R nQ .. , , 2 , 1.6
p2 .
13
-
1.2 , , supremum.
, , - . ( R) R nQ ( R).YKNOTHTA . a; b a < b x a < x < b.
. a < b. c, c =
p2: a c < b c,
r a c < r < b c. r + c a < r + c < b.
- .
- - .
1.1. y > 0,m; k 2 Z, n; l 2 N, n; l 2 mn = kl . ( npy)m = ( l
py)k.
. c = ( npy)m d = ( lpy)k, c = d., ,
cnk = ( npy)mnk =
( npy)nmk
= ymk; dml = ( lpy)kml =
( lpy)lmk
= ymk
, ,cnk = dml: (1.3)
, nk = ml. nk = ml 6= 0, (1.3) c = d. nk = ml = 0 , , m = k = 0, ( npy)m = ( lpy)k = 1 .
. , y > 0 r 2 Q r . m 2 Z, n 2 N n 2 r = mn . , 1.1, ( n
py)m
. ,
yr = ( npy)m;
m 2 Z, n 2 N n 2 r = mn ., r 2 Q r > 0,
0r = 0:
n 2 N n 2 y1/n = npy y 0. .
, yr > 0 y > 0 r. 1.4.5 .
, , . .
, yr - r - y , , r 0, yr y.
14
-
.
, x y > 1.
X = fyr j r 2 Q; r < xg:
X - r < x ( , r = [x]) yr X . X . , r 2 Q r < x r < [x] + 1 , , yr < y[x]+1( ). y[x]+1 X . supX .
. 10 y > 1 x ,
yx = supfyr j r 2 Q; r < xg:
, y = 1 x ,
1x = 1:
0 < y < 1 x , 1y > 1, (1y )
x
yx = 1(1/y)x :
, x x > 0,
0x = 0:
, , . y < 0, yx x .11 y = 0, yx x . y > 0, yx x.
1.2. y > 1. b > 1 n 2 N 1 < y1/n < b.. y > 1 b > 1, , b1y1 > 0, n 2 N 0 < 1n < b1y1 . (1.2) , n,
bn 1 n(b 1) > y 1, , y1/n < b. 1 < y1/n .
1.3. x 2 Q y > 1. yx = supfyr j r 2 Q; r < xg.. x 2 Q y > 1. r 2 Q r < x, yr < yx. yx fyr j r 2 Q; r < xg. a fyr j r 2 Q; r < xg. , a > 0. ( ) a < yx, y
x
a > 1. , 1.2, n 2 N 1 < y1/n < yxa , , a < yx(1/n). , , x 1n 2 Q x 1n < x , , yx(1/n) fyr j r 2 Q; r < xg. yx a. yx fyr j r 2 Q; r < xg.
10 2.4.20.11, 4.3, yx - y < 0.
15
-
1.3 yx = supfyr j r 2 Q; r < xg x . , x : yx.
1.4. [] r 2 Q r < x1 + x2. r1; r2 2 Q r1 < x1 r2 < x2 r1 + r2 = r.[] x1; x2 > 0, r 2 Q 0 < r < x1x2. r1; r2 2 Q 0 < r1 < x1 0 < r2 < x2 r1r2 = r.
. [] r 2 Q r < x1 + x2. r x1 < x2, r2 2 Q r x1 < r2 < x2. r1 = r r2, r1 2 Q. , r1 < x1 r2 < x2 r1 + r2 = r.[] x1; x2 > 0, r 2 Q 0 < r < x1x2. 0 < rx1 < x2, r2 2 Q 0 < rx1 < r2 < x2. r1 = rr2 , r1 2 Q. , 0 < r1 < x1 0 < r2 < x2 r1r2 = r.
1.8 . - .
1.8. [] y; y1; y2 > 0. y1xy2x = (y1y2)x, yx1yx2 = yx1+x2 , (yx1)x2 =yx1x2 .[] x1 < x2. y > 1, yx1 < yx2 . 0 < y < 1, yx1 > yx2 .[] 0 < y1 < y2. x > 0, y1x < y2x. x < 0, y1x > y2x.
. [] .(i) y > 1, yr yx r 2 Q r < x.( yx fyr j r 2 Q; r < xg.)(ii) y > 1 yr a r 2 Q r < x, yx a.(, , a fyr j r 2 Q; r < xg yx .) . y1; y2 > 1. r 2 Q r < x. (y1y2)r = y1ry2r y1xy2x. (y1y2)x y1xy2x. r1; r2 2 Q r1; r2 < x. r = maxfr1; r2g 2 Q, r < x, y1r1y2r2 y1ry2r = (y1y2)r (y1y2)x. y1r1 (y1y2)
x
y2r2. y1x (y1y2)
x
y2r2.
y2r2 (y1y2)x
y1x, , y2x (y1y2)
x
y1x. y1xy2x (y1y2)x.
(y1y2)x y1xy2x y1xy2x (y1y2)x y1xy2x = (y1y2)x. . . y > 1. r1; r2 2 Q r1 < x1 r2 < x2. yr1yr2 = yr1+r2 yx1+x2, , yr1 yx1+x2yr2 . yx1 y
x1+x2
yr2 . yr2 yx1+x2yx1 , ,
yx2 yx1+x2yx1 . yx1yx2 yx1+x2 . r 2 Q r < x1 + x2. r1; r2 2 Q r1 < x1 r2 < x2 r1 + r2 = r. yr = yr1+r2 = yr1yr2 yx1yx2 . yx1+x2 yx1yx2 . yx1yx2 yx1+x2 yx1+x2 yx1yx2 yx1yx2 = yx1+x2 . . . y > 1 x1; x2 > 0. r1; r2 2 Q r1 < x1 r2 < x2. r01; r02 2 Q r1 r01
16
-
r2 r02 0 < r01 < x1 0 < r02 < x2. (yr01)r
02 = yr
01r02 yx1x2 , ,
yr1 yr01 (yx1x2)1/r02 . yx1 (yx1x2)1/r02 . yx1 yr01 > 1, (yx1)r2 (yx1)r
02 yx1x2 , , (yx1)x2 yx1x2 .
r 2 Q r < x1x2. r0 2 Q r r0 0 < r0 < x1x2. r1; r2 2 Q 0 < r1 < x1 0 < r2 < x2 r1r2 = r0. yr yr0 = yr1r2 =(yr1)r2 (yx1)r2 (yx1)x2 . yx1x2 (yx1)x2 . (yx1)x2 yx1x2 yx1x2 (yx1)x2 (yx1)x2 = yx1x2 . .[] x1 < x2 y > 1. r1; r2 2 Q x1 < r1 < r2 < x2. r 2 Q r < x1 yr yr1 . yx1 yr1 . , yr1 < yr2 yx2 . yx1 < yx2 . 0 < y < 1 .[] 0 < y1 < y2 x > 0. 1 < y2y1 , [], 1 = (
y2y1)0 < (y2y1 )
x , [], y1x < y1x(y2y1 )x = (y1
y2y1)x = y2
x. x < 0 .
. . y > 1, yx x R
, 0 < y < 1, yx x R. , . x > 0, yx -
y [0;+1) , x < 0, yx y (0;+1)..
a+1 = +1 a > 1; a+1 = 0 0 a < 1;a1 = 0 a > 1; a1 = +1 0 < a < 1;
(+1)b = +1 b > 0 b = +1; (+1)b = 0 b < 0 b = 1:
00; 1+1; 11; (+1)0; 01
.
1.1. , , , .
.
, , .
1.3. a > 0, a 6= 1. y > 0 x ax = y.. , a > 1 y > 1.
X = fx j ax yg:, 0 2 X . , 1.2, n 2 N an > y. , x 2 X ax y an, x n. X - , supX .
= supX:
17
-
a = y. a < y. 1.2 n 2 N a1/n < y
a. a+(1/n) < y,
+ 1n 2 X . X . a y. a > y. 1.2, , n 2 N a1/n < ay . y < a(1/n), x 2 X ax y < a(1/n) , , x < 1n . 1n X . X . a y. a y a y a = y. . a > 1 y = 1, ax = y x = 0. a > 1 0 < y < 1, , 1y > 1, a
= 1y , , = a = a = y., 0 < a < 1 y > 0, , 1a > 1, (
1a)
= y, = a = a = y. ax = y : a1 = y a2 = y, a1 = a2 , 1 = 2.
. 12 y > 0 a > 0, a 6= 1, ax = y, - 1.3, y a
loga y:
a > 1. , , y = ax (1;+1) (1;+1). y = ax ., 1.3 (0;+1). , y = ax (1;+1) (0;+1). , - , x = loga y (0;+1) (1;+1). y = ax x (1;+1), x = loga y y (0;+1).
0 < a < 1, a > 1 . y = ax x (1;+1) (0;+1) x = loga y y (0;+1) (1;+1). , .
1.9 .
1.9. a; b > 0, a; b 6= 1.[] loga(y1y2) = loga y1 + loga y2 y1; y2 > 0.[] loga(yz) = z loga y y > 0 z.[] logb y =
loga yloga b
y > 0.[] loga 1 = 0, loga a = 1.[] 0 < y1 < y2. a > 1, loga y1 < loga y2. 0 < a < 1, loga y1 > loga y2.
. [] x1 = loga y1 x2 = loga y2, ax1 = y1 ax2 = y2. ax1+x2 = ax1ax2 = y1y2, loga(y1y2) = x1 + x2 = loga y1 + loga y2.[] x = loga y, ax = y. azx = (ax)z = yz , , loga(yz) = zx =z loga y.[] x = logb y w = loga b, bx = y aw = b. awx = (aw)x = bx = y. loga y = wx = loga b logb y.[] loga 1 = 0 a0 = 1 loga a = 1 a1 = a.
12 2.4.21.
18
-
[] x1 = loga y1 x2 = loga y2, y1 = ax1 y2 = ax2 . ax1 < ax2 , a > 1, x1 < x2 , 0 < a < 1, x1 > x2.
.
1.4.1. infimum supremum (a; b) \ (R nQ) = fx 2 R nQ j a < x < bg.1.4.2. a A = fr 2 Q j r < ag. A Q A - Q. , u 2 Q (, u = [a] + 1) r u r 2 A. Q, A. Q supremum.
1.4.3. [] a; b > 0 m;n 2 Z, anbn = (ab)n, aman = am+n, (am)n = amn.[] m;n 2 Z m < n. a > 1, am < an. 0 < a < 1, an < am.[] 0 < a < b n 2 Z. n > 0, 0 < an < bn. n < 0, 0 < bn < an.
1.4.4. n;m 2 N n;m 2.[] y; y1; y2 0, npy1y2 = npy1 npy2 n
pmpy = nm
py.
[] n < m. y > 1, mpy < npy. 0 < y < 1, npy < mpy.[] 0 y1 < y2. npy1 < npy2.1.4.5. r; r1; r2 2 Q.[] y; y1; y2 > 0, y1ry2r = (y1y2)r, yr1yr2 = yr1+r2 , (yr1)r2 = yr1r2 .[] r1 < r2. y > 1, yr1 < yr2 . 0 < y < 1, yr1 > yr2 .[] 0 < y1 < y2. r > 0, y1r < y2r. r < 0, y1r > y2r.
1.4.6. y > 1. yx = inffyr j r 2 Q; x < rg.1.4.7.
p3, 7p129 3
p2 +
p5 .
1.4.8. 13 [] n; k;m 2 N k;m > 1. npk/m k;m n- .
[] a0; a1; : : : ; an 2 Z k;m 2 Z > 1. km anxn + an1xn1 + + a0, k a0 m an.
1.4.9. m;n 2 N, m;n 2. , m;n, logm n .
13 1.6.
19
-
20
-
2
.
2.1 .
. ( ) x : N! R N . n 2 N x(n) , , xn: , n 2 N
xn = x(n):
, , - : x1, x2, x3 . , / -. xn+1 xn xn1 xn: n, N, - . x : N ! R ,,
(x1; x2; : : : ; xn; : : : ) (xn) (xn)+1n=1:
, , , x, n, : (yn), (xk), (zm) .
, , . , n,m, k, , - ( 0), n 2 N m 2 N k 2 N. 2.1.1. ( 1n) (1;
12 ;
13 ; : : : ;
1n ; : : : ).
2.1.2. (n) (1; 2; 3; 4; : : : ; n; : : : ).
2.1.3. (1) (1; 1; 1; : : : ; 1; : : : ).
2.1.4. ((1)n1) (1;1; 1;1; : : : ; 1;1; : : : ). 2.1.5.
1
10n
( 110 ;
1102; 1103; : : : ; 110n ; : : : ).
2.1.6. n- n, (1; 2; 2; 3; 2; 4; 2; 4; 3; 4; 2; : : : ). n- .
2.1.7. (m n)+1n=1 (m 1;m 2;m 3; : : : ;m n; : : : ). 2.1.8. (m n)+1m=1 (1 n; 2 n; 3 n; : : : ;m n; : : : ).
21
-
- (xn)+1n=1; (xn), , , .
: . -, . (1)+1n=1 f1g . , , (1; 1; 1; : : : ). , ., . , (1;1; 1;1; 1;1; : : : ), (1; 1;1; 1; 1;1; 1; 1;1; : : : ) f1; 1g. (1; 12 ; 13 ; 14 ; 15 ; 16 ; : : : ), (12 ; 1; 14 ; 13 ; 16 ; 15 ; : : : ) f 1n jn 2 Ng.. (xn) xn+1 xn n, - xn+1 > xn n, xn+1 xn n xn+1 < xn n. (xn) . (xn) , c xn = c n. , (c) (c; c; c; : : : ; c; : : : ).
. 2.1.1, 2.1.5, 2.1.7 , 2.1.2, 2.1.8 , 2.1.3 2.1.4, 2.1.6 .
. (xn) , u : xn u n. u (xn). (xn) -, l : l xn n. l (xn). (xn) , - l u : l xn u n.
, u (xn), u0 u , l (xn), l0 l . 2.1.9. (c) .
2.1.10. ( 1n), (1)n1
n
, (n1n ), ((1)n1)
[1; 1].
2.1.11. (1+(1)n1)n
2
(1; 0; 3; 0; 5; 0; 7; 0; : : : ) -
. , l 0 . , , . .
2.1.12. (1; 0;3; 0;5; 0;7; 0; : : : ), -, . l , l, l, .
22
-
2.1.13. ((1)n1n) (1;2; 3;4; 5;6; : : : ) - . , , - , , .
(xn) , l; u
l xn u
n. , xn 2 [l; u] n. , , 0 [M;M ] [l; u],
M xn M
, , jxnj M n. , (xn) , M jxnj M n. : M jxnj M n, M xn M n, M M (xn). : (xn) M jxnj M n.
n. : n 234 4 n n2 n > 8 xn < xn+1 (xn).
. , n, , , n n0 n n0:
n (xn) (xn), (xn) , , n . : xn+1 xn; xn+1 > xn;xn u, xn = c. , (xn) , , , , u, c.
2.1.14. (1; 23 ; 7;2;1;1;1;1; : : : ) , .
2.1.15. (n2 14n + 8) . , n2 14n+8 = (n 7)2 41 n 7 , , .
n n00, n000 n n00 n n000.
n0 = maxfn00; n000g:
n0 n00, n n0. , n0 n000, n n0. n n0. : , , , .
2.1.16. n2 3n 37 n 8. , 2n+1n+1 > 2513 n 13. n2 3n 37 2n+1n+1 > 2513 n maxf8; 13g = 13.
23
-
, , - : , n00; n000; , . ,, !
. n .
2.1.17. (1)n1 > 0 n, n. , (1)n1 0 n, n.
n n . , n , , , n.
.
2.1.1. , .
2.1.2. a+b2 + (1)n1 ab2
.
m 2 N, n m[ nm ]+1n=1. m = 1; 2; 3.
2.1.3. : 1; 4; 9; 16, 25; 36. 49; 24; ;
2.1.4. - .
2.1.5. (xn) (yn) (xn + yn). , , . , , . (xn) (yn) (xnyn). , , , xn; yn 0 n.
2.1.6. ((1)n1n), (1)n1n , 18nn2+n+1, 13nn! , n302n , 2[n2 ], n 3[n3 ] ; ; ;
2.1.7. n k n > k . n .
2.1.8. , , , , . , - .
2.1.9. (xn) . c xn = c n.
2.1.10. 1 a, b, p, q, p, q 0, (xn) x1 = a x2 = b xn+2 = pxn+1 + qxn.
1 n- .
24
-
(i) p 6= 0, q = 0. xn = bpn2 n 2.(ii) p = 0, q 6= 0. xn = aq n12 n xn = bq n22 n.(iii) p 6= 0, q 6= 0. x2 = px + q , 1, 2, , xn = 1
n1 + 2n1 n. x2 = px + q , , , xn =
n1 + (n 1)n1 n. x2 = px+ q , , , 2 [0; 2) xn = n1 cos(n 1) + n1 sin(n 1) n. n- 2 x1 = x2 = 1 xn+2 = 3xn, xn+2 = xn+1 + xn, xn+2 =2xn+1 xn, xn+2 = xn+1 xn.2.1.11. ( ) (xn) xn+1 = x1 + + xn ; xn+3 = xnxn+2xn+1 , xn+1 = 1 1xn xn+1 = 2 1xn .
2.2 , .
2.2.1. ( 1n) :
1; 12 ;13 ;
14 ;
15 ; : : : ;
1100 ;
1101 ; : : : ;
1100000 ; : : : ;
1100000000 ; : : : :
, n , 1n , , 1n 0.
2.2.2. (n1n ) :
0; 12 ;23 ;
34 ;
45 ; : : : ;
99100 ;
100101 ; : : : ;
99999100000 ; : : : ;
99999999100000000 ; : : : :
, n , n1n 1 . , n1n 1 jn1n 1j = 1n , , n , 1n .
(xn) : n , xn x .. xn x jxnxj . n - n . ,, : jxnxj n .
:
2.2.1. ( 1n). 1n 0 j 1n 0j = 1n , , - n .
2 Fibonnaci 1; 1; 2; 3; 5; 8; 13.
25
-
; , 0:000132. 1n 0:000132 n - ; , : n 1n 0:000132; : 1n < 0:000132 n >
1000000132 = 7575:75 : : : .
n > 1000000132 ; 7576 >1000000132 , , n -
7576, 1n < 0:000132. , 0:0000000000132. , n - 75757575758, 1n < 0:0000000000132. , , : ( 0:000132 0:0000000000132) ( 7576 75757575758). , , . - . , , . , , , n0; , n n0; 1n < . , , - : > 0 n0 1n0 ,, 1n0 ;
1n0+1
; 1n0+2 ; : : : < . ( ) n0 . n0 ( ) 1n < ; , . 1n < n >
1 (, n)
.
2.1. a 0, n0 = [a] + 1 n > a. a < 0, n0 = 1 n > a.
. .
: n > 3 1, n > 83 3 = [83 ] + 1 n > 2 3 = 2 + 1 = [2] + 1. ( 1 0) n0 n0 = [1 ] + 1. , ( ) n0 , n0:
, .
. (xn) x (xn) x x (xn) > 0 n0 jxn xj < n n0:: (xn) x > 0 jxn xj < . (xn) x
xn ! x limxn = x limn!+1 xn = x: (xn) , (xn) .
, , (xn) x n- xn x n .
:
xn ! x , 8 > 0 9n0 2 N 8n 2 N (n n0 ) jxn xj < )
26
-
xn ! x > 0 n0 ( )
n n0 ()) jxn xj <
, ,
jxn xj < (() n n0:
jxn xj < n n0: : n . :
jxn xj < ( P1 ( ( Pm ( n n0; P1; : : : ; Pm . : - . , -. . .
2.2.1 .
2.2.1.
1n ! 0:
> 0. n0 j 1n 0j < n n0: , , j 1n 0j < n n0:, j 1n 0j < ( ) 1n < ( ) n > 1 . :
j 1n 0j < ( 1n < ( n > 1 :
, 1.1 n0 n0 > 1 : , - n > 1 n0 n n0: n0 n n0 n > 1 j 1n 0j < . :
n n0 ) n > 1 ) 1n < ) j 1n 0j < :
( ) n0: , 1 0, n0 = [1 ] + 1 n >
1 . , n0
n0, j 1n 0j < n n0: 2.2.3. (c) c.
c! c:
> 0. n0 jc cj < n n0: jccj < ( ) 0 < . , 0 < n ( n). , n0 ( , n0 = 1), jc cj < n n0: 2.2.4. ((1)n1) . ( ) ((1)n1) x. > 0 n0 n n0 j(1)n1xj < . ,
27
-
n0; n n0:, n n0 j1xj < n n0 j1 xj < . j 1 xj < 1 2 (x ; x+ ) j1xj < 1 2 (x ; x+ ). , > 0 1, 1 (x ; x+ ). , , 0 < 1, 2, . 2.2.5. (1)
n1n ! 0.
> 0. (1)n1
n 0 < ( ) 1n <
( ) n > 1 : , n0 >1 :
n0; n n0 n > 1 j 1n 0j < . 2.2.6. n2+1
2n3n ! 0. > 0.
n2+12n3n 0
< ( ) n2+12n3n <
( ) 2n3 n > 1 (n2 + 1): , n, . ( )
n2+12n3n n
2+n2
2n3n3 =2n ;
n2+12n3n < (, )
2n < -
( ) n > 2 . . n0 > 2 : n0; n n0 n > 2
n2+12n3n 0
< . n2+1
2n3n , ,2n : -
, , :
a b; a < b < : a < b < . , b a a.
. (xn) +1 (xn) +1 +1 (xn) M > 0 n0 xn > M n n0: : (xn) +1 M > 0 xn > M . (xn) +1,
xn ! +1 limxn = +1 limn!+1 xn = +1:, (xn) 1 (xn) 1 1 (xn) M > 0 n0 xn < M n n0: : (xn) 1 M > 0 xn < M . (xn) 1,
xn ! 1 limxn = 1 limn!+1 xn = 1: , , : (xn) +1 n-
xn +1 n . 1.
:
xn ! +1 , 8M > 0 9n0 2 N 8n 2 N (n n0 ) xn > M)
xn ! 1 , 8M > 0 9n0 2 N 8n 2 N (n n0 ) xn < M)
28
-
2.2.7.
n! +1:
M > 0. n0 n > M n n0: . 1.1, n0 > M . , n > M n0 n n0. n0 n n0 n > M . ( ) n0 = [M ] + 1 , n0 n0, n > M n n0. 2.2.8. ((1)n1n) (1;2; 3;4; 5;6; : : : ) +1 1. ( ) +1. M > 0 n0 (1)n1n > M n n0: , n0; n n0: n n0 n > M n n0 n > M . , n > M , , -, . , 1 . 2.2.9. n3+n
n2+3! +1.
M > 0. n3+nn2+3
> M n3+n > Mn2+3M . n, n3+n
n2+3> M n3+n
n2+3
. ( )
n3+nn2+3
n3n2+3n2
= n4 :
n3+nn2+3
> M n4 > M n > 4M . - n0 n0 > 4M , , n0; n n0 n > 4M n3+n
n2+3> M .
n3+nn2+3
, , n4 : , :
a b; a > M b > M:
a > M b > M . - , b a a.3
!, lim, limn!+1 , 1. - , 1. limn!+1 xn; , R.
, .
. > 0. (x ; x+ ) - x. (1 ;+1] - +1 [1;1 ) - 1.
Nx() = (x ; x+ ); N+1() = (1 ;+1]; N1() = [1;1 ):3 2.2.6.
29
-
M = 1 , =1M
+1 (1 ;+1] (M;+1] 1 [1;1 ) [1;M).
: x 2 R , Nx() . x Nx(). Nx() x Nx()., . l < x, x ( +1) l, > 0 Nx() l., x < u, x ( 1) u, > 0 Nx() u., , l < x < u, x () l; u, > 0 Nx() l; u.
jxn xj < xn ! x , ,x < xn < x+ , , xn 2 (x ; x+ ) , , xn 2 Nx()., xn > M xn < M xn ! 1, , xn 2 (M;+1] xn 2 [1;M) , , xn 2 N+1() xn 2 N1(), = 1M :4
.
. x 2 R: xn ! x > 0 n0 xn 2 Nx() n n0 , , > 0 xn 2 Nx().
, , - . , , 1, - .5
.
2.2.1. , : 1n+8 ! 0, 3n+12n+5 ! 32 ; 1pn+5 ! 0,2n2n! +1, n
2n21 ! 0, n2 7n! +1, n4n2+1n2+3
! +1, 2n 2n/2 ! +1, 15n4n ! 0.2.2.2. .
2.2.3. x; y 2 R; x 6= y. > 0 Nx() \Ny() = ;.2.2.4. x 2 R: > 0 n Nx( 1n) Nx().
T>0Nx() = fxg , ,
x x.
T+1n=1Nx(
1n) = fxg.
2.2.5. xn ! x. > 0 n0() n0 xn 2 Nx() n n0. , 0 < 0 < , n0(0) n0().2.2.6. 6 (xn) x > 0 xn /2 Nx() n.
4 , xn +1 1 xn 2 N+1() = ( 1 ;+1] xn 2 N1() = [1; 1 ).
5 , .
6 .
30
-
2.2.7. 7 0 > 0. xn ! x 0 < 0 xn 2 Nx().2.2.8. (xn) x : n0 > 0 jxn xj < n n0. xn ! x. (xn) .
2.2.9. [] (xn) . , xn ! x, (xn) x .[] (xn) xn 2 Z n. xn ! x, (xn) x 2 Z.2.2.10. x (xn).[] , > 0 jxn xj < , jxn xj . . : xn = (1)n1, x = 0, = 1. : xn ! x > 0 jxn xj .[] , M > 0 xn > M , M xn M . . : xn = 1,M = 1. : xn ! +1 M > 0 xn M .
2.3 .
2.1 , , .
2.1. . , .
. (xn), (yn) k0;m0
xk0 = ym0 ; xk0+1 = ym0+1; xk0+2 = ym0+2; : : : : (2.1)
xn ! a 2 R. > 0. xn 2 Na() . (2.1) yn 2 Na() . , xn 2 Na() n0 . . . n0 k0, xn 2 Na() k0 , , yn 2 Na() m0 . . n0 > k0, n0 = k0 + p yn 2 Na() m0 + p . yn ! a. 2.3.1. (1; 12 ;
13 ;
14 ;
15 ;
16 ; : : : ), (2; 5; 14 ; 15 ; 16 ; : : : ). -
0. , - . 0.
7 . > 0 0 > 0.
31
-
2.3.2. (xn) (x1; x2; x3; : : : ). (xn+1) (x2; x3; x4; : : : ) (xn+2) (x3; x4; x5; : : : ). , (xn+m) (x1+m; x2+m; x3+m; : : : ). 2.1,
limn!+1 xn = x limn!+1 xn+m = x:
: 1n+3 ! 0 1n ! 0.
.
2.2 : .
2.2. xn ! x 2 R u, l.[] x > u, xn > u.[] x < l, xn < l.[] u < x < l, u < xn < l.
. [] > 0 Nx() u. xn 2 Nx(), xn > u.[] > 0 Nx() l. xn 2 Nx(), xn < l.[] > 0 Nx() u; l. xn 2 Nx(), u < xn < l. . [] [] xn > u xn < l. xn > u xn < l , , u < xn < l.
2.3 . .
2.3. .
. (xn) a; b. > 0 Na() Nb() . xn 2 Na() , , xn 2 Nb(). xn 2 Na() xn 2 Nb() . . (xn) a; b c a; b. [] [] 2.2 xn < c xn > c. xn < c xn > c .
2.4. [] xn ! x 2 R, (xn) .[] xn ! +1, (xn) .[] xn ! 1, (xn) .. [] x, (x1; x+1). (xn) n0 (x1; x+1). , (xn) (x1; x+1) x1; : : : ; xn01. , , , (x 1; x + 1) [l; u] (xn). (xn) .[] +1, (1;+1]. (xn) n0 (1;+1]. (xn) (1;+1] x1; : : : ; xn01.
32
-
, , , (1;+1] [l;+1] (xn). (xn) ., l xn > l. l (xn), (xn) .[] [].
2.3.3. H ((1)n1) . - 2.4[].
2.3.4. (1+(1)n1)n
2
, (1; 0; 3; 0; 5; 0; 7; : : : ),
. , +1, , 2.2, > 1, ., (1; 0;3; 0;5; 0;7; : : : ) , 1. 2.4[,] .
.
2.5 : . 2.5 2.2. , . 2.5 - : - , .8
2.5. [] xn l n xn ! x 2 R, x l.[] xn u n xn ! x 2 R, x u.[] u < l xn u n xn l n, (xn) .. [] x < l, , 2.2, xn < l, xn l n . x l.[] .[] (xn) , , [] [], u l , , l u . (xn) . 2.3.5. xn ! x 2 R xn 2 [l; u] n, x 2 [l; u]. 2.3.6. ((1)n1n), (1;2; 3;4; 5;6; : : : ), , 1 1. 2.3.7.
n 3[n3 ]
, (1; 2; 0; 1; 2; 0; 1; 2; 0; : : : ), ,
2 0. 2.6. xn yn n xn ! x 2 R yn ! y 2 R, x y.. ( ) y < x. a y < a < x. yn < a a < xn. , yn < a a < xn. yn < xn , , xn yn n. . x y.
8 . 2.5.15.
33
-
2.3.8. 1n < 1n n 1n ! 0 1n ! 0. , xn < yn n xn ! x 2 R yn ! y 2R, x < y. xn < yn n xn yn n x y. x < y x = y .
2.7 2.8 - : ( ) .
2.7. xn yn.[] xn ! +1, yn ! +1.[] yn ! 1, xn ! 1.. [] M > 0. xn > M , yn xn, xn > M yn xn. yn > M . ,yn ! +1.[] .
2.3.9. 2n + (1)nn = 2n n n n n ! +1. ,2n+ (1)nn! +1. 2.3.10. n2+2n+1n+2 n n n! +1. n
2+2n+1n+2 ! +1.
2.3.11. n! n n. n!! +1. 2.8 .
2.8. xn yn zn. xn ! a zn ! a, yn ! a.. > 0. xn 2 (a ; a + ) , , zn 2(a ; a + ). xn; zn 2 (a ; a + ) , yn xn; zn, yn 2 (a ; a+ ). yn ! a.
2.3.12. 1n (1)n
n 1n n. 1n ! 0 1n ! 0, - (1)
n
n ! 0.
2.3.13. 0 n2[n/2]n 1n n. 0! 0 1n ! 0, n2[n/2]
n ! 0.
.
.
. (xn) (xn). (xn) (yn) (xn + yn). (xn) (yn) (xn yn). xn yn =xn + (yn), (xn) (yn) (xn) (yn). (xn) (yn) (xnyn). (xn) (xn). , (xn) () (xn) (xn)
1xn
.
34
-
1xn
xn 6= 0 n.
(xn) (yn) xnyn
. xnyn = xn
1yn;
(xn) (yn) (xn) (yn).
xnyn
yn 6= 0 n.
(xn) (jxnj). .
2.9. x; y 2 R .[] xn ! x, xn ! x.[] xn ! x, jxnj ! jxj.[] xn ! x yn ! y x+ y , xn + yn ! x+ y.[] xn ! x yn ! y x y , xn yn ! x y.[] xn ! x yn ! y xy , xnyn ! xy.[] xn ! x x , xn ! x.[] xn 6= 0 n. xn ! x 1x (, x 6= 0), 1xn ! 1x :[] yn 6= 0 n. xn ! x yn ! y xy , xnyn ! xy :. [] xn ! x x 2 R. > 0 jxn xj < ,,
j(xn) (x)j = jxn xj < : xn ! x. xn ! +1. M > 0 xn > M , , xn < M . xn ! 1, xn ! (+1). xn ! 1. M > 0 xn < M , , xn > M . xn ! +1, xn ! (1).[] xn ! x x 2 R. > 0 jxn xj < , ,jxnj jxj jxn xj < : jxnj ! jxj. xn ! +1. M > 0 xn > M , , jxnj > M . jxnj ! +1, jxnj ! j+1j. xn ! 1. M > 0 xn < M , , jxnj > M . jxnj ! +1, jxnj ! j 1j.[] xn ! x yn ! y x; y 2 R. > 0 jxn xj < 2, , jyn yj < 2 , ,
j(xn + yn) (x+ y)j = j(xn x) + (yn y)j jxn xj+ jyn yj < 2 + 2 = :
xn + yn ! x+ y. xn ! +1 yn ! y y 2 (1;+1]. (yn) , l yn > l n. , M > 0 xn > M l, ,
xn + yn > (M l) + l =M:
xn + yn ! +1, xn + yn ! x+ y. [].
35
-
[] [] [].[] xn ! x yn ! y x; y 2 R. (yn) , M 0 jynj M n. , > 0 jxn xj < 2M+1 ,, jyn yj < 2jxj+1 , ,
jxnyn xyj = j(xn x)yn + x(yn y)j jxn xjjynj+ jxjjyn yj M2M+1 + jxj2jxj+1 < : xnyn ! xy. xn ! +1 yn ! y y 2 (0;+1]. l 0 < l < y. l < yn. , M > 0 xn > Ml , ,
xnyn >Ml l =M:
xnyn ! +1, xnyn ! xy. [].[] [].[] xn ! x x 2 Rnf0g xn 6= 0 n. jxnj ! jxj. jxj > 0, l 0 < l < jxj jxnj > l. , > 0 jxn xj < jxjl , , 1
xn 1x
= jxnxjjxjjxnj < jxjljxjl = : 1xn ! 1x . xn ! +1. > 0 xn > 1 , , 1
xn 0 = 1xn < :
1xn ! 0, 1xn ! 1+1 . xn ! 1 [].[] [] [].
2.3.14. xn ! x, x 2 R k 2 N, xnk ! xk., xnk = xn xn (k ) ! x x (k ) = xk. , n1n ! 1 (n1n )3 ! 13 = 1. xn ! +1, , , xnk ! +1. , xn ! 1, xnk ! +1, k , xnk ! 1, k ., ( ), , xn ! 1, 1xnk ! 0.:
nk !(+1; k 2 Z, k > 00; k 2 Z, k < 0
2.3.15.(1)n1 = 1 ! 1 ((1)n1) . ,
2.9[] ., , x = 0, : xn ! 0 jxnj ! j0j = 0., jxn 0j <
jxnj 0 < xn ! 0 jxnj ! 0 . 2.3.16. a0 + a1x+ + akxk . -, k 1 ak 6= 0.
a0 + a1n+ + aknk = nka0
1nk
+ a11
nk1 + + ak1 1n + ak:
36
-
ak, 0. ,nk ! +1.
a0 + a1n+ + aknk ! ak(+1) =(+1; ak > 01; ak < 0
., limn!+1(a0 + a1n+ + aknk) = limn!+1 aknk. : 3n2 5n+ 2! +1 12n5 + 4n4 n3 ! 1.: 2n5 2n2 + n 7! 1, (2n5 2n2 + n 7)8 ! +1.: n3 + 2n 1! 1, (n3 + 2n 1)5 ! 1.
2.3.17. a0+a1x++akxk
b0+b1x++bmxm , ak 6= 0, bm 6= 0.
a0+a1n++aknkb0+b1n++bmnm =
nk
nm
a0
1nk
+ + ak1 1n + ak
b01nm + + bm1 1n + bm
:
ak bm. n
k
nm = nkm,
a0+a1n++aknkb0+b1n++bmnm !
8>:(ak/bm)(+1); k > mak/bm; k = m0; k < m
. : n32n2+n+1
2n23n1 ! +1, n2+n
n+2 ! 1, n4n3n4+1
! 1 n2+n+4n3+n2+5n+6
! 0.: 2n3+n2+n+12n+3 ! 1,
2n3+n2+n+12n+3
7 ! 1.: n3+n+73n3+n2+1 ! 13 ,
n3+n+73n3+n2+1
3 ! 127 . 2.3.18.
na !(+1; a > 00; a < 0
a > 0. M > 0 n0 na > M n n0:, na > M n > M1/a. 1.1, n0 > M
1/a. , n > M1/a n0 n n0. , n0, n > M1/a , , na > M n n0. ( ) n0 = [M1/a] + 1 , n0 n0, n > M1/a n n0. a < 0. , , na ! 0 , : a > 0 , ,na = 1
na ! 1+1 = 0. 2.3.19. (a; a2; a3; a4; : : : ), (an). : a. a = 1, (1) 1. , a = 0, (0) 0. a 1, (an) a 1, a2 1, a3 1, a4 1; : : : . , 1 1 , , .
37
-
a > 1. Bernoulli 1.1
an n(a 1) + 1
n. n(a 1) + 1! +1, an ! +1., , . M > 0 n0 an > M n n0. , an > M n > logaM . n0 > logaM , n0, n > logaM, , an > M n n0., 0 < jaj < 1. 1jaj > 1, janj = 1(1/jaj)n ! 1+1 = 0. an ! 0.:9
an
8>>>>>>>:! +1; a > 1! 1; a = 1! 0; 1 < a < 1 ; a 1
2.3.20. (1 + a + a2 + + an1 + an) a. :
1 + a+ a2 + + an8>:! +1; a 1! 1/(1 a); 1 < a < 1 ; a 1
a 1, 1 + a+ a2 + + an 1 + 1 + 1 + + 1 = n+ 1:
n+ 1! +1, 1 + a+ a2 + + an ! +1. . 1 < a < 1,
1 + a+ a2 + + an = an+11a1 ! 01a1 = 11a :
a 1. an+1 = 1 + (a 1)(1 + a + a2 + + an) n. 1+ a+ a2+ + an ! x 2 R, an+1 ! 1+ (a 1)x. , (an+1) . (1 + a+ a2 + + an1 + an) . 2.3.21.
loga n!(+1; a > 11; 0 < a < 1
a > 1. M > 0 n0 loga n > M n n0: a > 1, loga n > M n > aM . , n0 n0 > aM , , n0, loga n > M , ,n > aM n n0. 0 < a < 1. 1a > 1, loga n = log1/a n! (+1) = 1.
- - .9 2.4.9
2.3.22 2.3.25.
38
-
2.3.22.
npa! 1 a > 0:
a = 1 : np1 = 1! 1.
a > 1. Bernoulli (1+ a1n )n 1+na1n = a
, , 1 npa 1 + a1n n. , npa! 1.
0 < a < 1, 1a > 1, npa = 1
np
1/a! 11 = 1.
2.3.23.
npn! 1:
Bernoulli (1 +pn1n )
n 1 + npn1n =
pn ,
, 1 npn (1 +pn1n )
2 < (1 + 1pn)2 n. n
pn! 1.
2.10 .
2.10. 10 xn > 0 n.[] 0 < b < 1 xn+1xn b, xn ! 0.[] b > 1 xn+1xn b, xn ! +1.[] 0 a < 1 xn+1xn ! a, xn ! 0.[] a > 1 xn+1xn ! a, xn ! +1.. , .[] , n0 xn+1xn b n n0. n n0 + 1
0 < xn =xnxn1
xn1xn2
xn0+2xn0+1
xn0+1xn0
xn0 b b b b xn0 = bnn0 xn0 = xn0bn0 bn = c bn;
c = xn0bn0 . 0 < b < 1, bn ! 0. xn ! 0.
[] , n0 xn+1xn b n n0. n n0 + 1
xn =xnxn1
xn1xn2
xn0+2xn0+1
xn0+1xn0
xn0 b b b b xn0 = bnn0 xn0 = xn0bn0 bn = c bn;
c = xn0bn0 . b > 1, bn ! +1. xn ! +1.
[] b a < b < 1. xn+1xn b [] xn ! 0.[] b a > b > 1. xn+1xn b [] xn ! +1. 2.3.24.
an
nk! +1 a > 1 k 2 N:
an+1/(n+1)k
an/nk= a
n
n+1
k ! a a > 1. annk! +1.
: (an) a > 1 (nk).
10 2.4.11.
39
-
2.3.25.
an
n! ! 0:
a = 0, , , ann! = 0! 0. a 6= 0. jajn+1/(n+1)!jajn/n! = jajn+1 ! 0. jaj
n
n! ! 0 , , an
n! ! 0. : (an) a > 1 (n!).
2.11 2.9 .
2.11. [] xn ! +1 (yn) , xn + yn ! +1. xn ! 1 (yn) , xn + yn ! 1.[] xn ! 0 (yn) , xnyn ! 0.[] xn ! +1 (yn) , xnyn ! +1. xn ! 1 (yn) , xnyn ! 1.[] xn 6= 0 n. xn ! 0 (xn) , 1xn ! +1. xn ! 0 (xn) , 1xn ! 1.[] xn 6= 0 n. jxnj ! +1, 1xn ! 0.. [] xn ! +1 (yn) , l yn l n. M > 0 xn > Ml , ,xn + yn > (M l) + l =M . xn + yn ! +1. .[] xn ! 0 (yn) , M 0 jynj M n. > 0 jxnj < M+1 , ,
jxnynj = jxnjjynj MM+1 < :
xnyn ! 0.[] xn ! +1 (yn) , l > 0 yn l. M > 0 xn > Ml , , xnyn > Ml l =M . xnyn ! +1. .[] xn ! 0 xn > 0. M > 0 jxn 0j < 1M . 0 < xn < 1M , , 1xn > M . 1xn ! +1. .[] xn 6= 0 n jxnj ! +1. 1jxnj ! 0 , , 1xn ! 0.
2.3.26. (xn) , xnn ! 0. , 1+(1)
n1n ! 0. : n3[n/3]n ! 0 sinnn ! 0.
2.3.27. (1)n1n (1)n1n = n ! +1.
1(1)n1n ! 0. , 1(1)n1n = (1)
n1n ! 0.
(1)n1n . x = 0 2.9[];
10 .
2.3.28. (1)n1n ! 0,
(1)n1n
.
40
-
2.3.28;
(1)n1n
, ,
. (xn) , ( 1xn ) +1. , (xn) , ( 1xn ) 1. , (xn) , . xn ! 0 jxnj ! 0 1jxnj ! +1. 1jxnj 1, , , 1xn 1 n 1xn 1 n. ( 1xn ) .
, , : 10 0 0 .
0+ 0 0 0 , , 2.11[].
. 10+ = +1; 10 = 1:
1.1 : , , , .
, 2.12, 4.3. , ( ) . - 1.4 ab. 00, 1+1,11, (+1)0 01.. (xn) (yn)
xn
yn.
2.12. xn > 0 n. xn ! x 2 R yn ! y 2 R xy , xnyn ! xy. , xn ! 0 yn ! 1, xn
yn ! +1. 2.3.29. 2.12, n
pa! 1 a > 0.
, (a) ( 1n). a ! a 1n ! 0, n
pa = a1/n ! a0 = 1.
2.3.30. (an) a > 1 0 < a < 1 ,, 2.12. (a) (n). a ! a n ! +1, an ! a+1, +1, a > 1, 0, 0 < a < 1. 2.3.31. n
pn ! 1 2.12.
(n) ( 1n), n! +1 1n ! 0, (+1)0 .
01 , , , - 2.12 .
. (0+)1 = +1:
41
-
.
2.3.1. (n+1)27(n+3)79
(2n+1)106
,n3+(1)nn2+13n2+2(1)n1n
,n(n+1)
n+4 4n3
4n2+1
,
((1 n)5 + n4), ( n3+n+13n2+3n+1
)9,
2n+3n
2n+1+3n+1
, (pn+ 1pn), (pn2 + n+ 1pn2 + 1).
2.3.2. , , (1+ 2+22+ +2n), 1+ 12 + + 12n ,(1 2 + 22 + + (1)n2n), 27
37+ 2
8
38+ + 2n+6
3n+6
,2n
3n +2n+1
3n+1+ + 22n
32n
.
2.3.3. 212+1
313+1 n1n+1
,23123+1
33133+1
n31n3+1
.
2.3.4. x 6= 1 (xn1xn+1) . x (x2n1
x2n+1) .
2.3.5. x limn!+1 (x+1)2n
(2x+1)n ;
2.3.6. x 6= 1 xn 6= 1 n. xn ! x xn
1xn ! x1x .
2.3.7. 11 3+(1)n
2n ! 0, 3+(1)n
2n > 0 n 3+(1)n2n
.
(3(1)n1)n
2 ! +1 (3(1)n1)n
2
.
2.3.8. (xn) n2 2n < n2xn n2 + 3. : n + 1 2nxn n + 2xn + 3, n2 + nxn 15n n2xn
2 2n(n 1)xn + n2 2n 3 0.2.3.9. xn ! x 2 R yn ! y 2 R. x < y, xn < yn.2.3.10. xn ! x 2 R yn ! y 2 R x 6= y. jx yj > a, jxn ynj > a.2.3.11. , xn 2 [l; u] n xn ! x, x 2 [l; u]. x (xn), xn 2 (l; u) n; ;
2.3.12. 2.5[], -
2(1)n1
,1 + (1)
n12
n, (1)n1 + 10n3
,(1)n1 nn+1
.
2.3.13. 2n+ (1)n1n! +1, 22n+(1)n1n ! 0, 12 + (1)n14 n ! 0.2.3.14.
3n2n+1
n+2
! +1, [pn]pn! 1 n+2
3n2n+13n2n+1
n+2
! 1.2.3.15. (1 + 1n)
n2 ! +1, (1 1n2)n ! 1, (1 + 1
n2)n ! 1.
2.3.16. 12 [nx]!
8>:+1; x > 00; x = 01; x < 0
[nx] [ny]!
8>:+1; x > y0; x = y1; x < y
nx [ny]
8>>>>>>>:! +1; x > y! 0; x = y 2 Z! 1; x < y ; x = y 2 Q n Z
11 0 +1, - , , .
12 x = y 2 R nQ 2.7.18.
42
-
2.3.17. (3n)!(n!)3
! +1.
2.3.18. 0 a b c. npan + bn ! b, npan + bn + cn ! c.2.3.19. n
pn3 ! 1, npn4 + 3n2 + n+ 1! 1.
2.3.20. a, [a]+[2a]++[na]n2
! a2 .2.3.21. n
n2+1+ n
n2+2+ + n
n2+n! 1 1p
n2+1+ 1p
n2+2+ + 1p
n2+n! 1.
2.3.22. limm!+1limn!+1(cosm!x)2n
=
(1; x 2 Q0; x 2 R nQ
2.3.23. (xn) - : xn+1 = xn+2, xn+3 = xn3, xn+1 = xn22, xn+2 = xn2+2,xn+1 = xn
2 + 3, xn+2 = xn+1 + xn3 ;
2.3.24. xn ! x yn ! y, maxfxn; yng ! maxfx; yg minfxn; yng !minfx; yg.2.3.25. : n 1n =
1n + + 1n (n )! 0 + + 0 (n ) = n0 = 0.
: (1 + 1n)n = (1 + 1n) (1 + 1n) (n )! 1 1 (n ) = 1n = 1.
2.9[,];
2.3.26. x1 > 0 xn+1 x1 + + xn n 2 N. 0 < a < 2, xnan ! +1. a = 2 (2n).2.3.27. (xn), (yn) (xn + yn) . (xn), (yn) (xnyn) .
2.3.28. (xn + yn) (xn), (yn) , , , . (xnyn) (xn), (yn) , , , .
2.3.29. 13 (xn), (yn) xn ! +1, yn ! 1 (xn + yn) (i) c 2 R (ii) . (xn), (yn) xn ! 0, yn ! +1 (xnyn) (i) c 2 R (ii) . (xn), (yn) xn ! 0, yn ! 0 (xnyn ) (i) c 2 R (ii) . (xn), (yn) xn ! +1, yn ! +1 (xnyn ) (i) c 2 [0;+1] (ii) . (xnyn ) c 2 [1; 0);
2.3.30. 14 (xn), (yn) xn ! +1, yn ! 0 xn
yn(i)
c 2 [0;+1] (ii) . xnyn c 2 [1; 0); (xn), (yn) xn ! 0, yn ! 0
xn
yn(i) -
c 2 [0;+1] (ii) . xnyn c 2 [1; 0); (xn), (yn) xn ! 0, yn ! 1
xn
yn(i)
c 2 f+1;1g (ii) . xnyn c 2 R;13 .14 . 2.4.10.
43
-
2.3.31. (xn), (yn) xn; yn > 0 n, xn ! 0, yn ! +1 (xnyn) .
2.3.32. x (rn) rn ! x. x (tn) tn ! x. x (rn) (sn) rn ! x sn ! x. .2.3.33. - A. A supA A supA. infA. [0; 2], [0; 2), f2g, [0; 1][f2g supremum . , ( ) , , . , , , . N, Z, Q, f 1n jn 2 Ng, (0; 1] \ (R n Q), (0; 2) \ Q , supremum infimum .
2.3.34. - A u A. u = supA A u. infA l A.
2.3.35. - A. supA 2 A A supA. supA /2 A, A supA. infA.
2.3.36. k 2 N, k 2 xn 0 n. xn ! x, kpxn ! k
px.
xn ! +1, kpxn ! +1. 2.12.
2.3.37. jxnxmj 1 n;m n 6= m. jxnj ! +1. (xn) ; (n), (n), ((1)n1n).2.3.38. xn ! x xn x n, supfxn jn 2 Ng = x. x < y xn < y n xn ! x, supfxnjn 2 Ng < y.2.3.39. xn ! x, k inffxn jn kg x supfxn jn kg.2.3.40. xn ! x. (xn) k xk x. (xn) k xk x.2.3.41. 15 [] Cesro: xn ! x 2 R, x1++xnn ! x. xn = (1)n1 n, x1++xnn ! 0. xn = 1+(1)
n
2 n n, x1++xn
n ! +1. (xn) . - Cesro.
15 2.7.13.
44
-
[] an+1 an ! a 2 R, ann ! a.
an
n
loga nn
a > 1.
[] Cesro: 16 (xn), (yn) yn > 0 n y1 + + yn ! +1. xnyn ! l 2 R, x1++xny1++yn ! l.[]17 xn > 0 n xn ! x 2 [0;+1], npx1 xn ! x.[] an > 0 n an+1an ! a 2 [0;+1], n
pan ! a.
( npn),
npn!
np(2n)!/(n!)2
.
2.3.42. [] n kn 2 N n;1; : : : ; n;kn :(i) M jn;1j+ + jn;kn j M n,(ii) n;1 + + n;kn ! 1,(iii) k n;k ! 0 ( n;k = 0 n kn < k). , xn ! x, n;1x1 + + n;knxkn ! x.[] xn ! x, 12nx1 + 12n1x2 + + 122xn1 + 12xn ! x.
2.4 .
2.1 . , , , , 1. - : ((1)n1) ((1)n1n) .
2.1 .
2.1. 18 . :[] (xn) , limn!+1 xn = supfxn jn 2 Ng.: (xn) , +1, , . - .[] (xn) , limn!+1 xn = inffxn jn 2 Ng.: (xn) , 1, , . - .
. [] - fxn jn 2 Ng. ( ) supremum , ( ) supremum +1. (xn) .
supfxn jn 2 Ng = +1
xn ! +1.M > 0. M fxn jn 2 Ng, n0 xn0 > M . (xn) ,
xn xn0 > M16 Cesro yn = 1 n.17 4.3.3 [] -
.18 supremum 2.4.17.
45
-
n n0. xn ! +1. (xn) .
x = supfxn jn 2 Ng xn ! x. > 0. x < x, x fxn jn 2 Ng. n0 x < xn0 . (xn) ,
x < xn0 xn n n0. , x fxn jn 2 Ng,
xn x < x+ n.
x < xn < x+ n n0. xn ! x.[] .
2.1. (xn) - , , 2.1, (xn) , x, . , xn x n. , , (xn) , xn < x n., xn0 = x n0, ( ) x = xn0 xn x n n0 , , . -, (xn) , xn < x n. . : (xn) xn ! x, xn x n. , , (xn) (, , ), xn < x n. (xn) xn ! x, xn x n. , , (xn) (, , ), xn > x n.
2.1 . , , ( ) .19
2.4.1. (xn) x1 = 1
xn+1 =3xn+6xn+4
n 1:
x1 = 1; x2 =95 ; x3 =
5729 ; x4 =
345173 :
, , 4, . , ,
xn 0 n. , x1 = 1 0 , xn 0, , , xn+1 0.
19, , 2.3.23.
46
-
, - xn+1 xn :
xn+1 xn = 3xn+6xn+4 3xn1+6xn1+4 =
6(xnxn1)(xn+4)(xn1+4) :
xn+1xn n , x2x1 > 0, xn+1 xn > 0 n. (xn) . xn+1 > xn 3xn+6xn+4 > xn , ,xn
2 + xn 6 < 0 , , 3 < xn < 2 n. , (xn) , , ., , xn ! x. xn 0 n x 0 , ,
x = 3x+6x+4 :
x = 3 x = 2 , x 0, x = 2. , xn ! 2., (xn) , xn < 2 n. . : : xn 0 n (xn) ( ), (xn) . xn ! +1, , xn+1 = 3+(6/xn)1+(4/xn) n, +1 = 3, . (xn) , , xn ! 2. : . x1 = 1, x2 = 1:8, x3 = 1:9655::: x4 = 1:994219::: . (xn) 2, xn < 2 n. , , , (xn) 2.
.
2.4.2. 1 + 11! +
12! + + 1n!
.
xn = 1 + 11! +12! + + 1n! n.
xn+1 = 1 +11! +
12! + + 1n! + 1(n+1)! = xn + 1(n+1)! > xn
n, (xn) .,
n! 2n1
n. , n = 1 n 2 n! =1 2 3 n 1 2 2 2 = 2n1.,
xn 1 + 11 + 12 + + 12n1 = 1 + 1(1/2n)
1(1/2) < 1 +1
1(1/2) = 3:
n. (xn) , , .
2.4.3. (1 + 1n)
n , ,
.20
20 2.4.4.
47
-
an =1 + 1n
n n. Newton ( 1.1) x = 1n y = 1,
an = 1 +n1
1n +
n2
1n2
+ + nk 1nk + + nn 1nn= 1 + 11! +
12!(1 1n) + + 1k!(1 1n)(1 2n) (1 k1n )
+ + 1n!(1 1n) (1 n1n ):(2.2)
, , n+ 1,
an+1 = 1 +n+11
1
n+1 +n+12
1
(n+1)2+ + n+1k 1(n+1)k + + n+1n 1(n+1)n
+n+1n+1
1
(n+1)n+1
= 1 + 11! +12!(1 1n+1) + + 1k!(1 1n+1)(1 2n+1) (1 k1n+1)
+ + 1n!(1 1n+1) (1 n1n+1) + 1(n+1)!(1 1n+1) (1 n1n+1)(1 nn+1):
(2.3)
(2.2) (2.3). k 2 k n, k- (2.2) k- (2.3), n n+1. , (2.3) , k = n+ 1.
an < an+1
n, (an) . xn = 1 + 11! +
12! + + 1n! n, .
(xn) xn < 3 n. (2.2) > 0 < 1,
an 1 + 11! + 12! + + 1k! + + 1n! = xn < 3 (2.4) n. (an) , , ,, . a (an).
an ! a: , , k (2.2) , () k-,
an 1 + 11! + 12!(1 1n) + + 1k!(1 1n)(1 2n) (1 k1n ): n! +1, ( k)
a 1 + 11! + 12! + + 1k! = xk: , , k , , a xn n. (2.4)
an xn a n , an ! a,
xn ! a:. 2.4.2 2.4.3,
1 + 11! +
12! + + 1n!
(1 + 1n)
n .
e. ,
e = limn!+11 + 1n
n= limn!+1
1 + 11! +
12! + + 1n!
:
48
-
e
1 + 11! +
12! + + 1n!
(1 + 1n)
n. e
, e . ((1 + 1n)
n) , (1 + 1n)n < e n.
, 1 + 11! +12! + + 1n! < e n.
, 1+1 ( 1.4 2.3). xn ! 1 yn ! +1 :xnyn ! 1. : xnyn ! 1yn = 1! 1 . , , (1+ 1n)n ! e . , 1 + 1n ! 1 n! +1 (1 + 1n)n ! 1.. e - y > 0, loge y,
log y ln y:
2.13 , , 1.9.
2.13. [] log(y1y2) = log y1 + log y2 y1; y2 > 0.[] log(yz) = z log y y > 0 z.[] loga y =
log ylog a y > 0 a > 0, a 6= 1.
[] log 1 = 0, log e = 1.[] 0 < y1 < y2, log y1 < log y2.
.
2.4.4. 21
1 + 12 +13 + + 1n ! +1:
xn = 1 + 12 +13 + + 1n n.
xn+1 xn = 1n+1 > 0 n, (xn) , , . n
x2n xn = 1n+1 + + 1n+n 1n+n + + 1n+n = n 1n+n = 12 : (xn) x, , (xn) , xn x2n x x. xn ! x, x2n ! x. x2n xn ! x x = 0 , x2n xn 12 n. xn ! +1. xn ! +1 . , n k 0 2k n < 2k+1., n 2. , - k log2 n < k + 1, k = [log2 n]. , k,
xn = 1 +12 + (
13 +
14) + (
15 +
16 +
17 +
18) + + ( 12k1+1 + + 12k ) + 12k+1 + + 1n
1 + 12 + (14 + 14) + (18 + 18 + 18 + 18) + + ( 12k + + 12k )= 1 + 12 + 2
14 + 4
18 + + 2k1 12k
= 1 + 12 +12 +
12 + + 12 = 1 + k2 :
xn 1 + 12 [log2 n] > 1 + 12(log2 n 1) = 12 log2 n+ 12
n , log2 n! +1, xn ! +1.21 2.5.4, 2.6.2 7.3.20
8.2.7, 8.2.10 8.3.1. 2.4.6, 6.4.11.
49
-
2.4.5. 22
1 + 1
22+ 1
32+ + 1
n2
:
xn = 1 + 122 +132
+ + 1n2
n. xn+1xn = 1(n+1)2 > 0 n, (xn) .
xn 1 + 112 + 123 + + 1(n1)n = 1 + (11 12) + (12 13) + + ( 1n1 1n) = 2 1n < 2
n. (xn) , , .
. [a1; b1], [a2; b2]; : : : [an+1; bn+1] [an; bn] n. , (an) - (bn) an bn n. :(i) (an) (bn) .(ii) x an x bn n.(iii) x (ii) bn an ! 0. (iii), (an), (bn) x .
. (an) (bn) ,
a1 an bn b1 n, (an) , , ( b1 ) (bn) , , ( a1 ).
an ! a; bn ! b:
an bn n, a b. ,
an a b bn n. x 2 [a; b] an a x b bn n. , x an x bn n, a x b, x 2 [a; b]. x an x bn n - [a; b]. , x [a; b] , , a = b , , bn an ! 0. x x = a = b.
an x bn , , x 2 [an; bn]. , an x bn n x [an; bn], . : [an; bn] , T+1
n=1[an; bn] = [a; b];
a = limn!+1 an b = limn!+1 bn.22 2.6.1, 8.2.7 8.2.10 6.4.11, 7.3.20 8.2.1.
50
-
2.4.6. p 2 N, p 2. x 2 [0; 1),
0 x < 1: (2.5)
, x p 0; 1p
;1p ;
2p
; : : : ;
p1p ; 1
1p . x; kp x < k+1p k = 0; 1; : : : ; p1. , , ,k px < k+1 , , k = [px]. k 0; 1; : : : ; p 1 (2.5) 0 px < p, px 0; 1; : : : ; p 1. x1 = [px] ( k )
x1p x < x1p + 1p x1 2 f0; 1; : : : ; p 1g: (2.6)
, [x1p ;x1p +
1p),
1p , p
x1p ;
x1p +
1p2
;x1p +
1p2; x1p +
2p2
; : : : ;
x1p +
p1p2; x1p +
1p
1
p2. x ,
, x1p +kp2 x < x1p + k+1p2 k p2x px1 < k + 1,
k = [p2x px1]. (2.6) 0 p2x px1 < p , , p2x px1 0; 1; : : : ; p 1. x2 = [p2x px1] ( k )
x1p +
x2p2 x < x1p + x2p2 + 1p2 x1; x2 2 f0; 1; : : : ; p 1g:
. n-
x1p + + xnpn x < x1p + + xnpn + 1pn x1; : : : ; xn 2 f0; 1; : : : ; p1g: (2.7)
[x1p + + xnpn ; x1p + + xnpn + 1pn ), 1pn , p [x1p + + xnpn + kpn+1 ; x1p + + xnpn + k+1pn+1 ) k = 0; 1; : : : ; p 1, 1
pn+1. x
x1p + + xnpn + kpn+1 x < x1p + + xnpn + k+1pn+1 k pn+1x pnx1 pxn < k+1 k = [pn+1x pnx1 pxn]. (2.7) 0 pn+1x pnx1 pxn < p, pn+1x pnx1 pxn 0; 1; : : : ; p 1. xn+1 = [pn+1x pnx1 pxn] x1p + +xn+1pn+1 x < x1p + +xn+1pn+1+ 1pn+1 x1; : : : ; xn+1 2 f0; 1; : : : ; p1g:
, , x 2 [0; 1) : (xn), (sn) (tn). n :
sn =x1p + + xnpn ; tn = x1p + + xnpn + 1pn :
, n, xn 0; 1; : : : ; p 1
sn x < tn (2.8)
51
-
n. , [sn; tn) [sn+1; tn+1). , (sn) (tn) . :
sn+1 = sn +xn+1pn+1
sn ; tn+1 =tn 1pn
+xn+1pn+1
+ 1pn+1
tn 1pn + p1pn+1 + 1pn+1 = tn:, (sn) (tn) - . ,
tn sn = 1pn ! 0;
(sn), (tn) . ; (2.8)
sn ! x; tn ! x: (xn) : p 1. ( ) n0 xn = p 1 n n0. n n0
tn+1 =tn 1pn
+xn+1pn+1
+ 1pn+1
= tn 1pn + p1pn+1 + 1pn+1 = tn:
(tn) , tn ! x, tn = x. (2.8).
. (xn) p- x. (sn) p- ( ) x (tn) - p- x.
: p = 2 0; 1, p = 3 0; 1; 2, p = 10 0; 1; : : : ; 9 p = 16 0; 1; : : : ; 14; 15.
p- 8.
.
2.4.1. 2nn!nn
4nn!nn
.
2.4.2. 2.4.2, n 4 1+ 11! + 12! +13! xn 1 + 11! + 12! + 13! + 123 + + 12n1 2 < 3212 e 3512 < 3. e.
2.4.3. 23 k, (1 + kn)n ! ek ( n ! +1, )
k 2 N. (1 + kn)
n ! ek k 2 Z.2.4.4. 24
(1 + 1n)
n , , : -
(1 + 1n)n (1 + 1n+1)n+1 nn+1
n(n+2)(n+1)2
n+1=
n2+2nn2+2n+1
n+1 Bernoulli.
(1 + 1n)
n+1 .
23 2.5.3.24 e.
52
-
, , e. (k+1k )
k e (k+1k )k+1 k = 1; 2; : : : ; n 1 nnn! en1 n
n+1
n! n. 25
npn!n ! 1e 26 n
pn!! +1.
2.4.5. 1+ 12+ + 1n log2 n+1 n, 2.4.4.
2.4.6. 2.4.4 (an) (bn) an = 1 + 12 + +1
n1logn bn = 1+ 12+ + 1nlogn n. (an) , (bn) .27
2.4.7. 2.4.2 2.4.3 , t > 0,
1 + t1! +
t2
2! + + tn
n!
(1 + tn)
n .28
2.4.8. [] x1 = 1 xn+1 = xn + 1xn2 n. (xn) .[] x1 > 0 xn+1 =
p2xn n. ,
x1, (xn) .[] 7xn+1 = xn3 + 6 n. , x1, (xn) .[] x1 > 0 xn+1 = 6+6xn7+xn n. , x1, (xn) .[] xn+1 = 2 1xn n. x1 = 1, (xn) . x1 > 1, (xn) . x1 < 1 x1 6= k1k k 2 N, (xn) . x1 = k1k k 2 N;[] x1; x2 > 0 xn+2 = xn+1 + 2xn n. (xn+1xn ) .[] xn+1 = sinxn n. (xn) , , .
2.4.9. 29 [] a > 1, (an) , an+1 = aan, an ! +1. 0 < a < 1.[] a > 1 k 2 N, (an
nk) , -
an+1(n+1)k
= a nk
(n+1)kan
nk, an
nk! +1.
[] a > 1, ( npa) , -
2npa2
= npa, n
pa! 1. a = 1, 0 < a < 1;
[] ( npn)
npn! 1.
[] (ann! ) , an+1
(n+1)! =an
n!a
n+1 , an
n! ! 0.25 7.3.19.26 2.3.41 , , 8.3.7.27 Euler .
6.4.11 7.3.20.28 2.5.5.29 2.3.19 2.3.22 2.3.25.
53
-
2.4.10. 30 (xn), (yn) xn ! 1, yn ! +1 xn
yn(i)
c 2 [0;+1] (ii) . xnyn c 2 [1; 0); (xn), (yn) xn ! 1, yn ! 1
xn
yn(i)
c 2 [0;+1] (ii) . xnyn c 2 [1; 0);2.4.11. [] b , 1b+1+
1b+2+
1b+3+ + 1b+n ! +1.
[]31 (1 + a1) (1 + an) 1 + a1 + + an, a1; : : : ; an 0, (1 a1) (1 an) 1 a1 an, 0 a1; : : : ; an 1. b , limn!+1 (a+1)(a+2)(a+n)(b+1)(b+2)(b+n) , - a = b, a > b, a < b.[] xn > 0 n. c < 0 n
xn+1xn
1 c, xn ! 0. c > 0 n
xn+1xn
1 c, xn ! +1. d < 0 n
xn+1xn
1! d, xn ! 0. d > 0 n
xn+1xn
1! d, xn ! +1. 2.10. (2n)!
4n(n!)2! 0. -
2.10;
2.4.12. 32 y 0 k 2 N, k 2. (xn) x1 > 0 xn+1 = k1k xn +
1k
yxnk1
n. xn > 0 n., Bernoulli, xnk y n 2, , xn+1 xn n 2. (xn) , x = limn!+1 xn, xk = y x 0.2.4.13. xn+1 xn+xn+22 n.33, , (xn) . (xn xn+1) xn xn+1 ! 0. (xn) . (xn) . (xn) , , , .
2.4.14. 0 < a b. x1 = a y1 = b xn+1 =
pxnyn yn+1 = xn+yn2 n,
(xn) , (yn) , xn yn n (xn), (yn) , GA (a; b).34
w1 = a z1 = b wn+1 = 2wnznwn+zn zn+1 =pwnzn n,
(wn) , (zn) , wn zn n (wn), (zn) , HG (a; b).35
a; b;HG (a; b);GA (a; b) H (a; b) = 2aba+b , G (a; b) =
pab A (a; b) = a+b2 ;
30 . 2.3.30.31 (1 + a)n 1 + na Bernoulli a 0
1 a 0, .32 1.2.33 (xn) . n, -
.34 GA (a; b) - a, b.35 HG (a; b) - a, b.
54
-
2.4.15. xn = 135(2n1)246(2n) n. (nxn2)
(n+ 12)xn
2 .
.
2.4.16. 1.2.18.
2.4.17. . - , supremum.36
. 12n ! 0., - A. x1 2 A y1 A. [x1; y1] A A. x1+y12 A, x2 = x1, y2 =
x1+y12 , ,
x2 =x1+y1
2 , y2 = y1. [x2; y2] A A. , [x1; y1], [x2; y2]; : : : [xn+1; yn+1] [xn; yn] yn xn = y1x12n1 n [xn; yn] an 2 A un A. u xn ! u, yn ! u , , an ! u, un ! u. u A.
2.4.18. 37 I , . ( ) (an) I = fan jn 2 Ng. [x1; y1] I y1 x1 > 0 a1 /2 [x1; y1]. [x2; y2] [x1; y1] y2 x2 > 0 a2 /2 [x2; y2]. , [x1; y1], [x2; y2]; : : : [xn+1; yn+1] [xn; yn] an /2[xn; yn] n. 2 [xn; yn] , , 6= an n. I .
2.4.19. 38 1 , n 2, 2n 2n . pn qn , , . p2 = 4
p2 q2 = 8
pn+1 = 2pn2 +
4 pn24n
1/21/2 qn+1 = 4qn2 + 4 + qn24n 1/21 n 2. qn = pn
1 pn2
4n+1
1/2 n 2. (pn) , (qn) pn < qn n 2. (pn), (qn) .39
2.4.20. 40 x (rn) rn ! x. , p- x - 2.3.32. . y > 1 x . (rn) rn ! x (yrn) . ( ) (rn) rn ! x (yrn)
36 supremum : .37 A, R,
a : N! A A. an = a(n), A A = fan jn 2 Ng , , A . A , .
38 .39
, , , 2, . , pn 2 qn n 2. pn ! 2 qn ! 2.
40 .
55
-
(rn) rn ! x. yx = limn!+1 yrn (rn) rn ! x. y > 1 x . yx = limn!+1 yrn (rn) rn ! x. 0 y 1 x . yx - ( 1.4). 1.8 .
2.4.21. [] y > 0. r1; r2 r1 < r2 r1; r2 6= 0, yr11r1
yr21r2
. x1 < x2 x1; x2 6= 0, yx11x1
yx21x2
.[]41 y > 0.
n( npy 1) .
. log y = limn!+1 n( n
py 1).
log y = limn!+1 yxn1xn
(xn) xn ! 0 xn 6= 0 n. 2.13 ( []) . a > 0, a 6= 1 y > 0. loga y = log ylog a 1.9.2.4.22. , - ;
2.5 .
. (xn). n1, n2, n3, : : : , nk, : : : n n1 < n2 < < nk < nk+1 < . (xn). x1, x2, : : : , xn, : : : xn1 ; xn2 ; : : : ; xnk ; : : : . : xn1 , xn2 . , (xnk). , (xnk) (xn).
, n1 < n2 < < nk < nk+1 < , .
2.5.1. n1 = 2, n2 = 5, n3 = 6, n4 = 9, n5 = 13, (xn) x2; x5; x6; x9; x13., n1 = 2, n2 = 5, n3 = 6, n4 = 10, n5 = 8 (xn). x2; x5; x6; x10; x8 (xn) : x10 x8 (xn) ( x9) x10 x8 .
.
2.5.2. nk = 2k k, (xn), (x2k) (x2; x4; x6; x8; x10; : : : ).
2.5.3. nk = 2k1 k, (xn), (x2k1) (x1; x3; x5; x7; x9; : : : ).
41 .
56
-
2.5.4. nk = k k, (xk) (x1; x2; x3; x4; x5; : : : ). (xn) (xn).
2.5.5. nk = 2k1 k, (x2k1) (x1; x2; x4; x8; x16; : : : ).
(xnk) k. k - 1; 2; 3; : : : , nk (xn).
, (xn) (xn). , , (xn) (xn). , (xn), (xn) . - (xn) n1. , xn1 , (xn) n2. , xn1 xn2 , (xn) n3 . (xnk) (xn) (xn) .
2.2. nk 2 N nk < nk+1 k. nk k k.. n1 1 n1 2 N. nk k k. nk+1 > nk nk; nk+1 2 N, nk+1 nk + 1 , , nk+1 k + 1. nk k k.
2.2
nk ! +1: 2.14. , .
. xn ! x 2 R (xnk) (xn). xnk ! x. > 0. n0 xn 2 Nx() n n0. nk ! +1. k nk n0 , , xnk 2 Nx(). xnk ! x.
2.14 , , : , .42
2.5.6. ((1)n1) ., (1)(2k1)1 = 1! 1 (1)(2k)1 = 1! 1.
2.15 .
2.15. x 2 R x2k ! x x2k1 ! x. xn ! x.. > 0. x2k ! x, xn 2 Nx() n n0 . , x2k1 ! x, xn 2 Nx() n n00 . xn 2 Nx() n ( n0 n00) . xn ! x.
42 . 2.5.15.
57
-
2.5.7. 43
1 12 + 13 14 + + (1)n1 1n
:
xn = 1 12 + 13 14 + + (1)n1 1n n. x2k+2 x2k = 12k+1 12k+2 > 0 k. ,
x2k = 1 (12 13) (14 15) ( 12k2 12k1) 12k < 1
k, . (x2k) , , ., x2k+1 x2k1 = 12k + 12k+1 < 0 k. ,
x2k1 = (1 12) + (13 14) + + ( 12k3 12k2) + 12k1 > 0
k, . (x2k1) - , , .,
x2k x2k1 = 12k ! 0; (x2k), (x2k1) . (xn) . x (xn), :
x2 < x4 < : : : < x2k < x2k+2 < : : : < x < : : : < x2k+1 < x2k1 < : : : < x3 < x1:
(x2k) x , , (x2k1) x. , , [x2; x1],[x4; x3], [x6; x5]; : : : x - .
. . , ((1)n1) . , ((1)n1), , : 1 1. ((1)n1) . , .
BOLZANO - WEIERSTRASS. .
. 44 (xn) l; u l xn u n. (xn) , : , , . [l; u] [l; l+u2 ], [
l+u2 ; u]. ()
(xn) [l; u], (xn). [l1; u1]. [l1; u1] [l; u],u1 l1 = ul2 [l1; u1] (xn). (xn) ( ) [l1; u1]: xn1 2 [l1; u1]. [l1; u1] [l1; l1+u12 ], [
l1+u12 ; u1]. [l1; u1] -
(xn), 43 2.6.3 6.4.11 8.3.9.44 2.5.9.
58
-
(xn). [l2; u2] ( ). [l2; u2] [l1; u1], u2 l2 = u1l12 [l2; u2] (xn). (xn) ( ) [l2; u2]: xn2 2 [l2; u2]., , n2 > n1. , (xn) [l2; u2]. [l2; u2] [l2; l2+u22 ], [
l2+u22 ; u2]. [l2; u2] -
(xn), (xn). [l3; u3]. [l3; u3] [l2; u2],u3 l3 = u2l22 [l3; u3] (xn). (xn) ( ) [l3; u3]: xn3 2 [l3; u3]. ,, n3 > n2. . , , [lk; uk] k
[lk+1; uk+1] [lk; uk]; uk+1 lk+1 = uklk2 k. , xnk (xn) k
nk+1 > nk; xnk 2 [lk; uk]
k. uk+1 lk+1 = uklk2 k uk lk = ul2k k,
uk lk ! 0: , (lk), (uk) . lk ! x uk ! x. nk+1 > nk k, (xnk) - (xn) , lk xnk uk k, xnk ! x.
+1 . - . (1; 0; 3; 0; 5; 0; 7; : : : ) +1., , +1, +1: , (1; 3; 5; 7; : : : ). .
2.16. [] - +1.[] 1.. 45 [] (xn) . u (xn) > u. 46( ) u (xn) > u. (xn) n0 (1; u]. (xn) (1; u] x1; : : : ; xn01. , , , (1; u] (1; u0] (xn). (xn) . . (xn) +1, .
45 2.5.9.46 2.4[]
2.4[].
59
-
(xn) > 1. : xn1 > 1. (xn) > 2. : xn2 > 2., , n2 > n1. > 2. (xn) > 3. : xn3 > 3 n3 > n2. . xnk (xn) nk+1 > nk xnk > k k. (xnk) (xn) xnk ! +1.[] .
.
2.5.1. a < b < c < d. ( ) a, b, c d. .
2.5.2. (xn) : () ,() , , , . - (xnk) (xn) .
2.5.3. 2.3.36 2.4.3 (1 + pqn)
n ! qpep p 2 Z q 2 N q 2. , (1 + rn)n ! er r 2 Q. x. > 0. ex, r; s 2 Q r < x < s ex < er < es < ex + . , ex < (1 + rn)n < (1 + xn)n < (1 + sn)n < ex + (1 + xn)n ! ex.2.5.4. (xn) (xnk) xnk ! x 2 R, xn ! x. (xn) (xnk) xnk ! x 2 R, xn ! x.47 xn = 1 + 12 +
13 + + 1n n. 2.4.4
x2n xn 12 n. x2k k2 + 1 k , , xn ! +1.2.5.5. 2.4.7, , t 0, 1 + t1! +t2
2! + + tn
n!
(1 + tn)
n .
2.5.6. [] x1 > 0 xn+1 = 1+ 2xn n. (x2k), (x2k1) . (xn) .[] x1 > 0 xn+1 = 1 + 31+xn n. (x2k), (x2k1) . (xn) .[] 0 < p < 1 xn+2 = (1 p)xn+1 + pxn n. (x2k), (x2k1) . (xn) . xn n, yn = xn+1 xn n. (xn).
47 2.4.4. 2.6.2 7.3.20 - 8.2.7, 8.2.10 8.3.1 2.4.6 6.4.11.
60
-
2.5.7. a; b; x 2 R a 6= b. x2k ! a x2k1 ! b - (xnk) xnk ! x. (xnk) (x2k) (x2k1). (xnk) (x2k) (x2k1); x = a x = b.
2.5.8. [] x 2 R x3k ! x x3k1 ! x x3k2 ! x. - 2.15, xn ! x. . a; b; c; x 2 R, a 6= b, a 6= c, b 6= c. x3k ! a x3k1 ! b x3k2 ! c (xnk) xnk ! x. x = a x = b x = c.[] N - (xn). N, . - , (xn) . N . . n 2 N n = 2m1k m 2 N k 2 N. N : A(m) = f2m1k j k g m 2 N. A(1) , A(2) - , A(3) . , xn = 1k n = 2
m1k m 2 N k 2 N. (xn) : (x(m)k )
+1k=1 m 2 N, ,
m 2 N, x(m)k = 1k k 2 N. , 0. (xn) 0, (x2m13)+1m=1.
2.5.9. (xn). xn m > n xm < xn.48
(xn) , (xn). (xn) , (xn) . . Bolzano - Weierstrass.
2.5.10. (xn) n (xnk) k.
2.5.11. 49 , - .
2.5.12. (xn) x 2 R (xn) x. : (xn) (xn) .
2.5.13. xn < x n. supfxn jn 2 Ng = x (xn) x. ( 1n). supf 1n jn 2 Ng = 1. , ( 1n) 0, ( 1n), 1. - ;
2.5.14. (xn) (xnk). (xnk) (xn).
48 xn +1.49 2.1.9.
61
-
2.5.15. (xn) , . 2.14. (xn) , l, u u < l xn u n xn l n. 2.5[].2.5.16. [] xn ! x xn 6= x n. (xn) .[] (rn) , rn = pnqn pn 2 Z qn 2 N n. qn ! +1.[] x (rn) rn ! x rn = pnqn pn 2 Z qn 2 N n. qn ! +1 pn ! x(+1).
2.6 .
. (xn) Cauchy > 0 n0 jxn xmj < n;m n0. :
limn;m!+1(xn xm) = 0:
: (xn) Cauchy .
2.17. (xn) , Cauchy.
. xn ! x. > 0. n0 jxn xj < 2 n n0. , n m, jxm xj < 2 m n0.
jxn xmj = j(xn x) (xm x)j jxn xj+ jxm xj < 2 + 2 =
n;m n0. (xn) Cauchy.
Cauchy 2.17.
CAUCHY. (xn) Cauchy, .
. 50 (xn) Cauchy. n0 jxn xmj < 1 n;m n0. (m = n0), n n0 jxn xn0 j < 1,
jxnj = j(xn xn0) + xn0 j jxn xn0 j+ jxn0 j < 1 + jxn0 j:
, M = 1+ jxn0 j, (xn) [M;M ], (xn) . (xn) , , Bolzano - Weierstrass, - (xnk) .
xnk ! x: xn ! x. > 0. n0
jxn xmj < 2 n;m n0:50 2.6.6, 2.6.7 2.7.14.
62
-
n n0 , , . nk ! +1, nk n0 , ,
jxn xnk j < 2 : (2.9), xnk ! x,
jxnk xj < 2 : (2.10) (2.9) (2.10). k (2.9) (2.10) k
jxn xj = j(xn xnk) + (xnk x)j jxn xnk j+ jxnk xj < 2 + 2 = : jxn xj < n n0, xn ! x.
Cauchy 2.1. (xn) x (xn), jxn xj x, jxnxmj . , , Cauchy Cauchy.
2.6.1. (xn) xn = 1+ 122 +132
+ + 1n2
n. , ( 2.4.5.51) (xn) .m > n.
jxn xmj = 1(n+1)2 + 1(n+2)2 + + 1(m1)2 + 1m2< 1n(n+1) +
1(n+1)(n+2) + + 1(m2)(m1) + 1(m1)m
= ( 1n 1n+1) + ( 1n+1 1n+2) + + ( 1m2 1m1) + ( 1m1 1m) = 1n 1m < 1n :
, > 0. n0 1n0 < . m > n n0 jxn xmj < 1n 1n0 < . (xn) Cauchy , , .
Cauchy R.
. Cauchy .
Bolzano - Weierstrass, - , , supremum.
.
2.6.1. (xn), (yn) Cauchy. , , (xn+yn),(xnyn) Cauchy.
2.6.2. 52 xn = 1 + 12 + + 1n n, 2.4.4 x2n xn 12 n. (xn) Cauchy; xn ! +1.
2.6.3. 53 xn = 1 12 + 13 14 + + (1)n1n n.
jxn xmj = 1n+1 1n+2 + + (1)
mn1m
1n+1 n;m n < m. (xn) Cauchy .
51 8.2.7 8.2.10 6.4.11, 7.3.20 8.2.1.52 2.4.4. 2.5.4 7.3.20
8.2.7, 8.2.10 8.3.1 2.4.6 6.4.11.53 2.5.7. 6.4.11 8.3.9.
63
-
2.6.4. [] 0 < M < 1 jxnxn+1j cMn. n0 jxn xmj c Mn1M n;m n0 n < m. (xn) Cauchy. x (xn), jxn xj c Mn1M .[] 0 < M < 1 jxn+1 xn+2j M jxn xn+1j.54 (xn) Cauchy. x (xn), c 0 jxn xj c Mn1M .[] x1 > 0 xn+1 = 1 + 31+xn n. (xn) . n- xn .[] 0 < jj < 1 xn+1 = a + sinxn n. (xn) . n- xn .
2.6.5. . , - , supremum.55
2.4.17 . , -. 12n ! 0 , , (xn), (yn) Cauchy. - .
2.6.6. 56 Cauchy (xn). [a1; b1] b1 a1 < 1 xn 2 [a1; b1]. - [a2; b2] [a1; b1] b2 a2 < 12 xn 2 [a2; b2]. , [ak; bk] k [ak+1; bk+1] [ak; bk] bk ak < 1k k xn 2 [ak; bk] k. x x 2 [ak; bk] k , , xn ! x.2.6.7. 57 Cauchy (xn). ln = inffxk j k ng un = supfxk j k ng n ln un ln ln+1 un+1 un n. un ln ! 0 x ln ! x un ! x. ln xn un n, xn ! x.
2.7 .
. (xn) x 2 R. x (xn) (xn) x.
2.7.1. (xn) , , . , ( 1n) 0 (n) 1. 2.7.2. (xn) xn = (1)n1 n - , 1 1. (x2k) 1 (x2k1) 1.
54 (xn) . 0 M 1, .55 supremum -
.56 Cauchy.57 Cauchy.
64
-
x (xn). (xnk) (xn) x. (x2k) (x2k1) (xn), (xnk) (x2k) (x2k1). (xnk) (x2k), (xnk) (x2k) - , (xnk), x , (x2k),
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