Papadimitrakis analysis

, - .

i

.

- . (-) , Supremum , , . , - , . - . .

, , Supremum, . , - , Bolzano - Weierstrass , , - .

: - . ( ) , , , , , .

.1. . , n n. , .2. , , 10 - . , , .3. Riemann , , Darboux - . , Riemann . Riemann -, Darboux Darboux Riemann.4. . . : R ( ). Rd - . , , () ., .5. , -

iii

, . , ( , 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 , .

, , ., , , - .

2013.

iv

1 . 11.1 R R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Supremum infimum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Supremum. . . . . . . . . . . . . . . . . . . . . 91.4 , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 . 212.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 . . . . . . . . . . . . . . . . . . . . . . . 312.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.7 . . . . . . . . . . . . . . . . . . . . 64

3 . 693.1 , . . . . . . . . . . . . . . . . . . 693.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.3 . . . . . . . . . . . . . . . . . . . . . . 823.4 . . . . . . . . . . . . . . . . . . . . . . . . . 1003.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.6 Cauchy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4 . 1074.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.3 . . . . . . . . . . . . . . . . . . . . . . . . 1204.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.5 . . . . . . . . . . . . . . . . . . . . . . . . 1324.6 . . . . . . . . . . . . . . . . . . . . . . . . . 141

5 . 1455.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1695.5 . . . . . . . . . . . . . . . . . . . . . 1765.6 . . . . . . . . . . . . . . . . . . . . . . . 1915.7 Taylor, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1985.8 , . . . . . . . . . . . . . . . . . . . . . . . . 200

v

6 Riemann. 2076.1 Darboux. . . . . . . . . . . . . . . . . . . . . . . . 2076.2 . Darboux. . . . . . . . . . . . . . . . . . . . . . . 2116.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2186.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2206.5 . Riemann. . . . . . . . . . . . . . . . . . . . . . . 235

7 . 2437.1 , . . . . . . . . . . . . . . . . . . . . . . 2437.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2497.3 . . . . . . . . . . . . . . . . . . . . . . . 2577.4 Taylor, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

8 . 2798.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2798.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . 2858.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2988.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 3078.5 Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3128.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

9 . 3199.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3199.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3219.3 Weierstrass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

10 . 33710.1 . . . . . . . . . . . . . . . . . . . . . . 33710.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34610.3 Taylor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36010.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 366

11 . 37311.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37311.2 , , . . . . . . . . . . . . . . . . . . . . . 37811.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 38911.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39511.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39911.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40311.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

12 . 42512.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42512.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43212.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43612.4 . . . . . . . . . . 44212.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

13 . 45713.1 Peano. . . . . . . . . . . . . . . . . . . . . . . 45713.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46313.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46913.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

vi

13.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

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1

.

1.1 R R.

R. , - , ,N, Z Q. : N = {1, 2, 3, . . . }. , 0.

- R .1

.

1.1. n 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 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 N. x, y

(x+ y)n =n

k=0

(nk

)xkynk =

nk=0

n!k!(nk)!x

kynk.

1 11 .2 Bernoulli 1.1.5 5.4.6.

1

. 3

(x+ y)n = (x+ y) (x+ y) (n )

, n x y . , k Z 0 k n xkynk x k y n k . , k Z 0 k n, xkynk n k k x. , k n (

nk

)= n!k!(nk)! =

n(n1)(nk+1)k! .

k Z 0 k n (nk

) xkynk.

, , |x| x - x 0. , , |x y| 4 x y. |x y| x y.

, .

1.2. [] a > 0, :

|x| a a x a.

|x| < a a < x < a.

[] x, y :

|x y| |x|+ |y|.

. x y 0 0. , .

. R,

R = R {,+}.

, + , +. :

< x, x < +, < +.

(+) = , () = + .

(+) + x = +, x+ (+) = +, (+) + (+) = +,3 Newton 1.1.5 5.4.8. -

, .4 .

, , d- . , 9 10. 11 .

2

() + x = , x+ () = , () + () = .

,

(+) + (), () + (+)

.

(+) x = +, x () = +, (+) () = +,

() x = , x (+) = , () (+) = .

(+) (+), () ()

.

()x = , x() = x > 0,

()x = , x() = x < 0,

()() = +, ()() = .

()0, 0()

.

1+ = 0,

1 = 0.

10

.

x = x > 0,

x = x < 0,

x = 0.

x0 ,

0 ,

,

.,5

|+| = +, | | = +.5 1.8,

2.3.28 2.12.

3

R R (a,+), [a,+), (, b), (, b] (,+) {x |x > a}, {x |x a}, {x |x < b}, {x |x b} R, -. R , , (a,+], [a,+], [, b), [, b],[,+), (,+], [,+].

R R . - ( ) ( ) . -, (+)+ (+) = +. , , - . (+) (+) . , ( ) , , . (+)0 , , .

a A , , . , , -, , a (3,+] a +. A [, 2] A .

.

1.1.1. 6 a x b a y b, |xy| ba .

1.1.2. x y < 0 z w < 0, 0 < yw xz.

1.1.3. [] b1, . . . , bn > 0. l a1b1 , . . . ,anbn

u, l a1++anb1++bn u.[]7 1, . . . , n 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 : |x+1| > 2, |x1| < |x+1|, xx+2 >

x+33x+1 ,

(x 2)2 4, |x2 7x| > x2 7x, (x1)(x+4)(x7)(x+5) > 0,(x1)(x3)

(x2)2 0. x x : (, 3], (2,+), (3, 7),(,2) (1, 4) (7,+), [2, 4] [6,+), [1, 4) (4, 8], (,2] [1, 4) [7,+).

1.1.5. Bernoulli Newton .6 , .7 y1, . . . , yn, yk k , -

,

1y1++nyn1++n

= 11++n

y1 + + n1++n yn = w1y1 + + wnyn,

wk = k1++n yk, yk y1, . . . , yn. , w1, . . . , wn > 0 w1 + + wn = 1, w1, . . . , wn w1y1 + + wnyn y1, . . . yn .

4

1.2 Supremum infimum.

. - A. A u x u x A. u A. A l l x x A. l A., A , - l u l x u x A.

: u A, u u A , l A, l 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 a. , , 1.3 l a. (a, b], (a, b), (a,+) l a ., , [a, b], (a, b], [a, b), (a, b), (a,+), [a,+) , a, l a . (, a].

1.2.2. , u b [a, b], [a, b), (a, b], (a, b), (, b), (, b]. , , - . [a, b], (a, b], (, b] 1.3 [a, b), (a, b), (, b). , , , b, u b . [b,+).

1.2.3. (a,+), [a,+), (,+) (, b), (, b], (,+) .

1.2.4. N , 1 , N l 1 . , 1 N N (, 1]. N .

5

SUPREMUM. -, .

supremum R. , , .

supremum. R R N. , R N supremum, 12.

, supremum, infimum. : 0 , , .

INFIMUM. -, .

. -, A.

A = {x |x A}.

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) (, b], (, b) supremum, b, [a, b],(a, b), (a, b], [a, b), (a,+), [a,+) infimum, a.

. , , A minimum A. , , A maximum A. minimum maximum A , ,

minA maxA.

[a, b] [a, b) -, - maximum maximum., -, supremum.

6

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 = {0} [2, 3] {4} minA = 0 maxA = 4. infA = 0 supA = 4.

1.2.7. minN = 1, infN = 1. , N : n N N n+ 1 N n., N N supremum. N . , , N - . .

1.2.8. A = { 1n |n N} = {1,12 ,

13 ,

14 , . . . } maxA = 1, supA = 1.

, A : 1n A A 1

n+1 A 1n . , A ,

, 0, A infimum. infA .

. - A , infA = . - A , supA = +.

(: ) . A , . , + , , , , , A.

- A supremum infimum. A -, supremum , , supremum +., A , infimum , -, infimum . , - A

infA supA.

, x A infA x x supA , , infA A supA A.

, supA, , +. , infA .

supA infA.

supA (i) (ii):(i) A supA. supA = + , supA , supA A.(ii) supA A., supA = + u, . u A ( A ), x A u < x. + A. supA > 0, . supA A ( supA A), x A supA < x , ( (i)),

supA < x supA.

7

x A, supA - . supA A.

, infA (i) (ii):(i) A infA.(ii) infA A.

supremum infimum .

A . A -: x1, x2 A x1 < x2 x x1 < x < x2 x A. : . 1.4 , - R, .

1.4. - A : x1, x2 A x1 < x2 x x1 < x < x2 x A. A .

. u = supA, l = infA,

l u +. , , A [l, u]. x (l, u). x A, x1, x2 A x1 < x < x2. , x 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. max{x, y} = x+y+|xy|2 min{x, y} =x+y|xy|

2 .

1.2.2. (a,+) [a, b) .

1.2.3. l a+ > 0, l a. |a b| > 0, a = b.

1.2.4. a 1 0 < < 1, a 0.

1.2.5. infA = infB supA = supB. A = B;

1.2.6. - A. [infA, supA] R A.

1.2.7. [a, b];

1.2.8. - A. ( ) supA A, : supA = + supA < +. A infA.

1.2.9. - A. supA A A . infA A.

1.2.10. A = [0, 2],A = [0, 2),A = [0, 1]{2}, ,, -, A u = supA : A (u , u] 6= > 0; A (u , u) 6= > 0; , , u / A; l = infA.

8

1.2.11. - A u. supA u x u x A. u supA . infA l.

1.2.12. - A,B.[] supA infB x y x A, y B.[] , . A = (, 0], B = [0,+) x y x A, y B. , x y x A, y B. A = (, 0], B = (0,+), A = (4,2), B = (2,+) A = (, 0), B = [1, 13]., , x y x A, y B. ( ) supA, infB x y x A,y B.[] x y x A, y B > 0 x A,y B y x . supA = infB x y x A, y B. ;[] 0 < x y x A, y B > 0 x A, y B yx 1 + . supA = infB x y x A, y B.

1.2.13. - A,B A B = R x < y x A,y B. A,B A B. A = (, ), B = [,+) A = (, ], B = (,+).

1.2.14. - A,B. A supA supB x A < x y B y > .

1.2.15. - A, B A B. infB infA supA supB.

1.2.16. [] - A, B. sup(A B) = max{supA, supB} inf(A B) = min{infA, infB}. sup(A B) min{supA, supB} inf(A B) max{infA, infB}.[] - A A = {x |x A}. sup(A) = infA inf(A) = supA.[] - A,B A+B = {x+ y |x A, y B}. 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 = {xy |x A, y B}. A B A = [3, 5],B = [1, 7], A = (3, 5),B = (1, 7) A = (1, 5),B = (2, 7) ; A,B (0,+), inf(A B) = infA infB sup(A B) = supA supB.

1.3 Supremum.

1.2.4 1.2.7.

1.1. N , , supN = +.

9

. ( ) N , supN . supN 1 N, n N supN 1 < n. supN < n+ 1 , n+ 1 N.

1.1 :

. l > 0, n N 0 < 1n < l.

, l > 0, 1l , N , n N 1l < n , , 0 0 A. , A 0.

1.5 .

1.5. x k Z k x < k + 1.

. x. 1.1 n N n > x m N m > x. l = m, l, n

l < x < n.

:

k Z, k x, k + 1 x.

k x k = l, k x k Z k l. , , k x k = n., , k Z k x k + 1 > x, k x < k + 1. k k x < k + 1 . k x < k+1 k x < k+1 k, k Z. k < k+1 k < k+1, 1 < k k < 1. k k Z, k k = 0, k = k.

. k Z k x < k+1, 1.5, x [x].

,[x] 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.

10

. b a > 0, , , n 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 N, l a. |a b| 1n n N, a = b.

1.3.2. infimum supremum {(1)nn |n N}, N { 1n |n N},{(1)n + 1n |n N}, {

1n |n N} {2

1n |n N},

+n=1[2n 1, 2n],

+n=1[

12n ,

12n1 ].

1.3.3. infimum supremum (a, b) Q = {x Q | a < x < b}.

1.3.4. 1.2.12. A = { 1n |n N}, B = {

1n |n N} x y

x A, y B. x y x A, y B. A = {r Q | r < 0}, B = {r Q | r > 0}.

1.3.5. r a r Q r < b, b a. {r Q | r > a} = {r Q | r > b}, a = b. {r Q | r < a} {r Q | r > b} = , a b. {r Q | r a} {r Q | r b} = Q, b a.

1.4 , , .

. n Z, n 1 (, n N), an :

an = a a (n ).

n Z, n 1 a 6= 0, :

an = 1aa (n ) .

, n = 0 a 6= 0, :a0 = 1.

1.4.3 . . .

., - , ,

. supremum.

1.2. n N, n 2. y 0 x 0 xn = y.

11

. 8 n N, n 2 y 0.

X = {x |x 0, xn y}.

, 0 X , X -. , y + 1 1 (y+1)n y+1 > y. x X xn y < (y+1)n , ,x < y + 1. X y + 1. X - , supX .

= supX.

, 0, 0 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, + X . , X . n y. n > y.

nynn1 > 0 > 0

ny

nn1 . , (1.2),

n ( )n nn1( ( )

)= nn1 n y.

y ( )n. x 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.

. n N, n 2. y 0, - xn = y, 1.2, n- y

ny.

n = 2, 2y y.

ny - y. , n- ( - xn = y) n

0 = 0

ny > 0 y > 0., - xn = y

y 0, . n , xn = y (i) , ny ny, y > 0, (ii) , 0, y = 0, (iii) , y < 0. n , xn = y (i) , ny, y > 0, (ii) , 0, y = 0, (iii) , n

y, y < 0. .

. ny. , supremum. .

8, 4.4.20 4 n- ny - y. 2.4.12.

12

1.4.4 , , . .

1.2 . , n N, xn (, , --) [0,+) [0,+). . 1.2 xn [0,+) [0,+). , y = xn , [0,+), , [0,+)., , x = ny [0,+) [0,+). , y = xn ( [0,+)) x = ny . , y = xn , x = ny .

- - . .

1.6. n, k. nk k n-

. , nk , .

. k n- , m N k = mn. n

k = m , , .

, nk , n

k = ml , m, l N. -

, , m, l > 1. n

k = 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 \Q .

. , , 2 , 1.6

2 .

1.2 , , supremum.

, , - . ( R) R \Q ( R).

YKNOTHTA . a, b a 0,m, k Z, n, l N, n, l 2 mn =kl . ( n

y)m = ( l

y)k.

13

. c = ( ny)m d = ( ly)k, c = d., ,

cnk = ( ny)mnk =

(( ny)n

)mk= ymk, dml = ( l

y)kml =

(( ly)l

)mk= ymk

, ,cnk = dml. (1.3)

, nk = ml. nk = ml 6= 0, (1.3) c = d. nk = ml = 0 , , m = k = 0, ( ny)m = ( ly)k = 1 .

. , y > 0 r Q r . m Z, n N n 2 r = mn . , 1.1, ( n

y)m

. ,

yr = ( ny)m,

m Z, n N n 2 r = mn ., r Q r > 0,

0r = 0.

n N n 2 y1/n = ny y 0. .

, yr > 0 y > 0 r. 1.4.5 .

, , . .

, yr - r - y , , r 0, yr y.

., x y > 1.

X = {yr | r Q, r < x}.

X - r < x ( , r = [x]) yr X . X . , r Q r < x r < [x] + 1 , , yr < y[x]+1( ). y[x]+1 X . supX .

. 9 y > 1 x ,

yx = sup{yr | r Q, r < x}.

, y = 1 x ,

1x = 1.

9 2.4.20.

14

0 < y < 1 x , 1y > 1, (1y )

x

yx = 1(1/y)x .

, x x > 0,

0x = 0.

, , . y < 0, yx x .10 y = 0, yx x . y > 0, yx x.

1.2. y > 1. b > 1 n N 1 < y1/n < b.

. y > 1 b > 1, , b1y1 > 0, n N 0 < 1n y 1

, , y1/n < b. 1 < y1/n .

1.3. x Q y > 1. yx = sup{yr | r Q, r < x}.

. x Q y > 1. r Q r < x, yr < yx. yx {yr | r Q, r < x}. a {yr | r Q, r < x}. , a > 0. ( ) a < yx, y

x

a > 1. , 1.2, n N 1 < y1/n < y

x

a , , a < yx(1/n). , ,

x 1n Q x1n < x , , y

x(1/n) {yr | r Q, r < x}. yx a. yx {yr | r Q, r < x}.

1.3 yx = sup{yr | r Q, r < x} x . , x : yx.

1.4. [] r Q r < x1 + x2. r1, r2 Q r1 < x1 r2 < x2 r1 + r2 = r.[] x1, x2 > 0, r Q 0 < r < x1x2. r1, r2 Q 0 < r1 < x1 0 < r2 < x2 r1r2 = r.

. [] r Q r < x1 + x2. r x1 < x2, r2 Q r x1 < r2 < x2. r1 = r r2, r1 Q. , r1 < x1 r2 < x2 r1 + r2 = r.[] x1, x2 > 0, r Q 0 < r < x1x2. 0 < rx1 < x2, r2 Q 0 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 Q r < x.( yx {yr | r Q, r < x}.)(ii) y > 1 yr a r Q r < x, yx a.(, , a {yr | r Q, r < x} yx .) . y1, y2 > 1. r Q r < x. (y1y2)r = y1ry2r y1xy2x. (y1y2)x y1xy2x. r1, r2 Q r1, r2 < x. r = max{r1, r2} 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 Q r1 < x1 r2 < x2. yr1yr2 = yr1+r2 yx1+x2, , yr1 y

x1+x2

yr2 . yx1 y

x1+x2

yr2 . yr2 y

x1+x2

yx1 , ,

yx2 yx1+x2

yx1 . yx1yx2 yx1+x2 .

r Q r < x1 + x2. r1, r2 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 Q r1 < x1 r2 < x2. r1, r2 Q r1 r1 r2 r2 0 < r1 < x1 0 < r2 < x2. (yr

1)r

2 = yr

1r

2 yx1x2 , ,

yr1 yr1 (yx1x2)1/r2 . yx1 (yx1x2)1/r2 . yx1 yr1 > 1, (yx1)r2 (yx1)r

2 yx1x2 , , (yx1)x2 yx1x2 .

r Q r < x1x2. r Q r r 0 < r < x1x2. r1, r2 Q 0 < r1 < x1 0 < r2 < x2 r1r2 = r. yr yr

= 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 Q x1 < r1 < r2 < x2. r 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

16

, 0 < y < 1, yx x R. , . x > 0, yx -

y [0,+) , x < 0, yx y (0,+).

.

a+ = + a > 1, a+ = 0 0 a < 1,

a = 0 a > 1, a = + 0 < a < 1,

(+)b = + b > 0 b = +, (+)b = 0 b < 0 b = .

00, 1+, 1, (+)0, 0

.

1.1. , , , .

., , .

1.3. a > 0, a 6= 1. y > 0 x ax = y.

. , a > 1 y > 1.

X = {x | ax y}.

, 0 X . , 1.2, n N an > y. , x X ax y an, x n. X - , supX .

= supX.

a = y. a < y. 1.2 n N a1/n < y

a. a+(1/n) < y,

+ 1n X . X . a y.

a > y. 1.2, , n N a1/n < ay . y < a(1/n),

x 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.

17

. 11 y > 0 a > 0, a 6= 1, ax = y, - 1.3, y a

loga y.

a > 1. , , y = ax (,+) (,+). y = ax ., 1.3 (0,+). , y = ax (,+) (0,+). , - , x = loga y (0,+) (,+). y = ax x (,+), x = loga y y (0,+).

0 < a < 1, a > 1 . y = ax x (,+) (0,+) x = loga y y (0,+) (,+). , .

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.[] 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 \Q) = {x R \Q | a < x < b}.

1.4.2. a A = {r Q | r < a}. A Q A - Q. , u Q (, u = [a] + 1) r u r A. Q, A. Q supremum.

1.4.3. [] a, b > 0 m,n Z, anbn = (ab)n, aman = am+n, (am)n = amn.[] m,n Z m < n. a > 1, am < an. 0 < a < 1, an < am.[] 0 < a 0, 0 < an < bn. n < 0, 0 < bn < an.

11 2.4.21.

18

1.4.4. n,m N n,m 2.[] y, y1, y2 0, n

y1y2 = n

y1 n

y2 n

my = nm

y.

[] n < m. y > 1, my < ny. 0 < y < 1, ny < my.[] 0 y1 < y2. n

y1 < n

y2.

1.4.5. r, r1, r2 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 = inf{yr | r Q, x < r}.

1.4.7. 3, 7

129 3

2 +

5 .

1.4.8. 12 [] n, k,m N k,m > 1. nk/m k,m n- .

[] a0, a1, . . . , an Z k,m Z > 1. km anxn + an1xn1 + + a0, k a0 m an.

1.4.9. m,n N, m,n 2. , m,n, logm n .

12 1.6.

19

2

.

2.1 .

. ( ) x : N R N . n N x(n) , , xn. , n N

xn = x(n).

, , - : x1, x2, x3 . , / -. xn+1 xn xn1 xn. n, N, - . x : N R ,,

(x1, x2, . . . , xn, . . . ) (xn) (xn)+n=1.

, , , x, n, : (yn), (xk), (zm) .

, , . , n,m, k, , - ( 0), n N m N k 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. (

110n

) ( 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)+n=1 (m 1,m 2,m 3, . . . ,m n, . . . ).

2.1.8. (m n)+m=1 (1 n, 2 n, 3 n, . . . ,m n, . . . ).

21

- (xn)+n=1, (xn), , , .

: . -, . (1)+n=1 {1} . , , (1, 1, 1, . . . ). , ., . , (1,1, 1,1, 1,1, . . . ), (1, 1,1, 1, 1,1, 1, 1,1, . . . ) {1, 1}. (1, 12 ,

13 ,

14 ,

15 ,

16 , . . . ), (

12 , 1,

14 ,

13 ,

16 ,

15 , . . . )

{ 1n |n N}.

. (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), u u , l (xn), l 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 [l, u] n. , , 0 [M,M ] [l, u],

M xn M

, , |xn| M n. , (xn) , M |xn| M n. : M |xn| M n, M xn M n, M M (xn). : (xn) M |xn| 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 n0, n0 n n0 n n0.

n0 = max{n0, n0}.

n0 n0, n n0. , n0 n0, 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 max{8, 13} = 13.

23

, , - : , n0, n0, , . ,, !

. 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 N, (n m[ nm ]

)+n=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)n1

n

),(

18nn2+n+1

),(13n

n!

),(n30

2n

),(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 , , , [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 |n1n 1| =

1n , , n ,

1n .

(xn) : n , xn x .. xn x |xnx| . n - n . ,, : |xnx| n .

:

2.2.1. ( 1n). 1n 0 |

1n 0| =

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 >1000000

132 , , 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 |xn x| < n n0.: (xn) x > 0 |xn x| < . (xn) x

xn x limxn = x limn+ xn = x.

(xn) , (xn) .

, , (xn) x n- xn x n .

:

xn x [ > 0 n0 N n N (n n0 |xn x| < )

]26

xn x > 0 n0 ( )

n n0 () |xn x| <

, ,

|xn x| < () n n0.

|xn x| < n n0. : n . :

|xn x| < P1 Pn n n0,

P1, . . . , Pn . : - . , -. . .

2.2.1 .

2.2.1.

1n 0.

> 0. n0 | 1n 0| < n n0. , , | 1n 0| < n n0., | 1n 0| < ( )

1n <

( ) n > 1 . :

| 1n 0| < 1n < n >

1 .

, 1.1 n0 n0 > 1 . , - n > 1 n0 n n0. n0 n n0 n > 1 |

1n 0| < . :

n n0 n > 1 1n < |

1n 0| < .

( ) n0. , 1 0, n0 = [1 ] + 1 n >

1 . , n0

n0, | 1n 0| < n n0.

2.2.3. (c) c.

c c.

> 0. n0 |c c| < n n0. |cc| < ( ) 0 < . , 0 < n ( n). , n0 ( , n0 = 1), |c c| < n n0.

2.2.4. ((1)n1) . ( ) ((1)n1) x. > 0 n0 n n0 |(1)n1x| < . ,

27

n0, n n0., n n0 |1x| < n n0 |1 x| < . | 1 x| < 1 (x , x+ ) |1x| < 1 (x , x+ ). , > 0 1, 1 (x , x+ ). , , 0 < 1, 2, .

2.2.5. (1)n1

n 0.

> 0. (1)n1

n 0 < ( ) 1n <

( ) n > 1 . , n0 >1 .

n0, n n0 n > 1 |1n 0| < .

2.2.6. n2+12n3n 0.

> 0. n2+12n3n 0

< ( ) n2+12n3n <

( ) 2n3 n > 1 (n2 + 1). ,

n, . ( )

n2+12n3n

n2+n2

2n3n3 =2n ,

n2+1

2n3n < (, ) 2n < -

( ) n > 2 . . n0 > 2 . n0, n n0 n >

2

n2+12n3n 0

< . n

2+12n3n , ,

2n . -

, , :

a b, a 0 n0 xn > M n n0. : (xn) + M > 0 xn > M . (xn) +,

xn + limxn = + limn+ xn = +.

, (xn) (xn) (xn) M > 0 n0 xn < M n n0. : (xn) M > 0 xn < M . (xn) ,

xn limxn = limn+ xn = .

, , : (xn) + n- xn + n . .

:

xn + [M > 0 n0 N n N (n n0 xn > M)

]xn

[M > 0 n0 N n N (n n0 xn < M)

]28

2.2.7.

n +.

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, . . . ) + . ( ) +. M > 0 n0 (1)n1n > M n n0. , n0, n n0. n n0 n > M n n0 n > M . , n > M , , -, . , .

2.2.9. n2+nn+3 +.

M > 0. n3+n

n2+3> M n3+n > Mn2+3M .

n, n3+n

n2+3> M n

3+nn2+3

. ( )

n3+nn2+3

n3n2+3n2

= n4 .

n3+n

n2+3> M n4 > M n > 4M . -

n0 n0 > 4M , , n0, n n0 n > 4M n

3+nn2+3

> M . n

3+nn2+3

, , n4 . , :

a b, a > M b > M.

a > M b > M . - , b a a.3

, lim, limn+ , . - , . limn+ xn, , R.

, .

. > 0. (x , x+ ) - x. (1 ,+] - + [,

1 ) - .

Nx() = (x , x+ ), N+() = (1 ,+], N() = [,1 ).

3 2.2.6.

29

M = 1 , =1M

+ (1 ,+] (M,+] [,1 ) [,M).

: x R , Nx() . x Nx(). Nx() x Nx()., . l < x, x ( +) l, > 0 Nx() l., x < u, x ( ) u, > 0 Nx() u., , l < x < u, x () l, u, > 0 Nx() l, u.

|xn x| < xn x , ,x < xn < x+ , , xn (x , x+ ) , , xn Nx()., xn > M xn < M xn , , xn (M,+] xn [,M) , , xn N+() xn N(), = 1M .

4

.

. x R. xn x > 0 n0 xn Nx() n n0 , , > 0 xn Nx().

, , - . , , , - .5

.

2.2.1. , : 1n+8 0,3n+12n+5

32 ,

1n+5

0,n2 7n +, n

2n21 0,n4n2+1n2+3

+, 2n 2n/2 +, 3+log2 n1+3 log2 n 13 .

2.2.2. .

2.2.3. x, y R, x 6= y. > 0 Nx() Ny() = .

2.2.4. x R. , 0 < 1 2, Nx(1) Nx(2). > 0 n Nx( 1n) Nx().

>0Nx() = {x} , ,

x x.

+n=1Nx(

1n) = {x}.

2.2.5. xn x. > 0 n0() n0 xn Nx() n n0. , 0 < < , n0() n0().

4 , xn + xn N+() = ( 1 ,+] xn N() = [,

1).

5 , .

30

2.2.6. 6 (xn) x > 0 xn / Nx() n.

2.2.7. 7 0 > 0. xn x 0 < 0 xn Nx().

2.2.8. (xn) x : n0 > 0 |xn x| < n n0. xn x. (xn) .

2.2.9. [] (xn) . , xn x, (xn) x .[] (xn) xn Z n. xn x, (xn) x Z.

2.3 .

2.1 , , .

2.1. . , .

. (xn), (yn) k0,m0

xk0 = ym0 , xk0+1 = ym0+1, xk0+2 = ym0+2, . . . . (2.1)

xn a R. > 0. xn Na() . (2.1) yn Na() . , xn Na() n0 . . . n0 k0, xn Na() k0 , , yn Na() m0 . . n0 > k0, n0 = k0 + p yn Na() m0 + p . yn a.

2.3.1. (1, 12 ,13 ,

14 ,

15 ,

16 , . . . ), (2, 5,

14 ,

15 ,

16 , . . . ). -

0. , - . 0.

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+ xn = x limn+ xn+m = x.

: 1n+3 0 1n 0.

6 .7 . > 0

0 > 0.

31

. 2.2 :

.

2.2. xn x R u, l.[] x > u, xn > u.[] x < l, xn < l.[] u < x < l, u < xn < l.

. [] > 0 Nx() u. xn Nx(), xn > u.[] > 0 Nx() l. xn Nx(), xn < l.[] > 0 Nx() u, l. xn 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 Na() , , xn Nb(). xn Na() xn Nb() . . (xn) a, b c a, b. [] [] 2.2 xn < c xn > c. xn < c xn > c .

2.4. [] xn x R, (xn) .[] xn +, (xn) .[] xn , (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,+]. (xn) n0 (1,+]. (xn) (1,+] x1, . . . , xn01. , , , (1,+] [l,+] (xn). (xn) ., l xn > l. l (xn), (xn) .[] [].

32

2.3.3. H ((1)n1) . - 2.4[].

2.3.4. ( (1+(1)n1)n

2

), (1, 0, 3, 0, 5, 0, 7, . . . ),

. , +, , 2.2, > 1, ., (1, 0,3, 0,5, 0,7, . . . ) , . 2.4[,] .

. 2.5 :

. 2.5 2.2. , . 2.5 - : - , .8

2.5. [] xn l n xn x R, x l.[] xn u n xn x 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 R xn [l, u] n, x [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 R yn y R, x y.

. ( ) y < x. a y < a < x. yn < a a < xn. , yn < a a < xn. yn < xn , , xn yn n. . x y.

2.3.8. 1n 0. xn > M , yn xn, xn > M yn xn. yn > M . ,yn +.[] .

2.3.9. 2n + (1)nn = 2n n n n n +. ,2n+ (1)nn +.

2.3.10. n2+2n+1n+2 n n n +. n2+2n+1

n+2 +.

2.3.11. n! n n. n! +.

2.8 .

2.8. xn yn zn. xn a zn a, yn a.

. > 0. xn (a , a + ) , , zn (a , a + ). xn, zn (a , a + ) , yn xn, zn, yn (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

).

(

1xn

) xn 6= 0 n.

(xn) (yn) (xnyn

). xnyn = xn

1yn,

(xn) (yn) (xn) (yn).

(xnyn

) yn 6= 0 n.

(xn) (|xn|).

34

.

2.9. x, y R .[] xn x, xn x.[] xn x, |xn| |x|.[] 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 R. > 0 |xn x| < ,,

|(xn) (x)| = |xn x| < .

xn x. xn +. M > 0 xn > M , , xn < M . xn , xn (+). xn . M > 0 xn < M , , xn > M . xn +, xn ().[] xn x x R. > 0 |xn x| < , ,|xn| |x| |xn x| < . |xn| |x|. xn +. M > 0 xn > M , , |xn| > M . |xn| +, |xn| |+|. xn . M > 0 xn < M , , |xn| > M . |xn| +, |xn| | |.[] xn x yn y x, y R. > 0 |xn x| < 2, , |yn y| < 2 , ,

|(xn + yn) (x+ y)| = |(xn x) + (yn y)| |xn x|+ |yn y| < 2 +2 = .

xn + yn x+ y. xn + yn y y (,+]. (yn) , l yn > l n. , M > 0 xn > M l, ,

xn + yn > (M l) + l =M.

xn + yn +, xn + yn x+ y. [].[] [] [].[] xn x yn y x, y R. (yn) , M 0 |yn| M n. , > 0 |xn x| < 2M+1 ,, |yn y| < 2|x|+1 , ,

|xnyn xy| = |(xn x)yn + x(yn y)| |xn x||yn|+ |x||yn y| M2M+1 +|x|

2|x|+1 < .

35

xnyn xy. xn + yn y y (0,+]. l 0 < l < y. l < yn. , M > 0 xn > Ml , ,

xnyn >Ml l =M.

xnyn +, xnyn xy. [].[] [].[] xn x x R\{0} xn 6= 0 n. |xn| |x|. |x| > 0, l 0 < l < |x| |xn| > l. , > 0 |xn x| < |x|l , , 1

xn 1x

= |xnx||x||xn| < |x|l|x|l = . 1xn

1x .

xn +. > 0 xn > 1 , , 1xn

0 = 1xn < .

1xn 0, 1xn

1+ . xn [].[] [] [].

2.3.14. xn x, x R k N, xnk xk., xnk = xn xn (k ) x x (k ) = xk. , n1n 1 (

n1n )

3 13 = 1. xn +, , , xnk +. , xn , xnk +, k , xnk , k ., ( ), , xn , 1xnk 0.:

nk

{+, k Z, k > 00, k Z, k < 0

2.3.15.(1)n1 = 1 1 ((1)n1) . ,

2.9[] ., , x = 0, : xn 0 |xn| |0| = 0., |xn 0| <

|xn| 0 < xn 0 |xn| 0 .

2.3.16. a0 + a1x+ + akxk . -, k 1 ak 6= 0.

a0 + a1n+ + aknk = nk(a0

1nk

+ a11

nk1+ + ak1 1n + ak

).

ak, 0. ,nk +.

a0 + a1n+ + aknk ak(+) =

{+, ak > 0, ak < 0

36

., limn+(a0 + a1n+ + aknk) = limn+ aknk. : 3n2 5n+ 2 + 12n

5 + 4n4 n3 .: 2n5 2n2 + n 7 , (2n5 2n2 + n 7)8 +.: n3 + 2n 1 , (n3 + 2n 1)5 .

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

(ak/bm)(+), k > mak/bm, k = m0, k < m

. : n

32n2+n+12n23n1 +,

n2+nn+2 ,

n4n3n4+1

1 n2+n+4n3+n2+5n+6

0.: 2n

3+n2+n+12n+3 ,

(2n3+n2+n+12n+3

)7 .: n

3+n+73n3+n2+1

13 ,

(n3+n+73n3+n2+1

)3 127 . 2.3.18.

na

{+, 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

+ = 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 , , . a > 1. Bernoulli 1.1

an n(a 1) + 1

n. n(a 1) + 1 +, an +., , . M > 0 n0

37

an > M n n0. , an > M n > logaM . n0 > logaM , n0, n > logaM, , an > M n n0., 0 < |a| < 1. 1|a| > 1, |an| = 1(1/|a|)n

1+ = 0. a

n 0.:9

an

+, a > 1 1, a = 1 0, 1 < a < 1 , a 1

2.3.20. (1 + a + a2 + + an1 + an) a. :

1 + a+ a2 + + an

+, 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 + a+ a2 + + an +. . 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 R, an+1 1+ (a 1)x. , (an+1) . (1 + a+ a2 + + an1 + an) .

2.3.21.

loga n

{+, a > 1, 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 (+) = .

- - .

2.3.22.

na 1 a > 0.

a = 1 : n1 = 1 1.

a > 1. Bernoulli (1+ a1n )n 1+na1n = a

, , 1 na 1 + a1n n. , n

a 1.

0 < a < 1, 1a > 1, na = 1

n

1/a 11 = 1.

9 2.4.9 2.3.22 2.3.25.

38

2.3.23.

nn 1.

Bernoulli (1 +n1n )

n 1 + nn1n =

n ,

, 1 nn (1 +

n1n )

2 < (1 + 1n)2 n. n

n 1.

2.10 .

2.10. 10 xn > 0 n.[] 0 1 xn+1xn b, xn +.[] 0 a < 1 xn+1xn a, xn 0.[] a > 1 xn+1xn a, xn +.

. , .[] , n0 xn+1xn b n n0. n n0 + 1

0 < xn =xn

xn1xn1xn2

xn0+2xn0+1xn0+1xn0

xn0 b b b b xn0 = bnn0 xn0 =xn0bn0 b

n = c bn,

c = xn0bn0 . 0 1, bn +. xn +.

[] b a b > 1. xn+1xn b [] xn +.

2.3.24.

an

nk + a > 1 k N.

an+1/(n+1)k

an/nk= a

(n

n+1

)k a a > 1. annk

+. : (an) a > 1 (nk).

2.3.25.

an

n! 0.

a = 0, , , an

n! = 0 0. a 6= 0. |a|

n+1/(n+1)!|a|n/n! =

|a|n+1 0.

|a|nn! 0 , ,

an

n! 0. : (an) a > 1 (n!).

10 2.4.11.

39

2.11 2.9 .

2.11. [] xn + (yn) , xn + yn +. xn (yn) , xn + yn .[] xn 0 (yn) , xnyn 0.[] xn + (yn) , xnyn +. xn (yn) , xnyn .[] xn 6= 0 n. xn 0 (xn) , 1xn +. xn 0 (xn) , 1xn .[] xn 6= 0 n. |xn| +, 1xn 0.

. [] xn + (yn) , l yn l n. M > 0 xn > Ml , ,xn + yn > (M l) + l =M . xn + yn +. .[] xn 0 (yn) , M 0 |yn| M n. > 0 |xn| < M+1 , ,

|xnyn| = |xn||yn| MM+1 < .

xnyn 0.[] xn + (yn) , l > 0 yn l. M > 0 xn > Ml , , xnyn > Ml l =M . xnyn +. .[] xn 0 xn > 0. M > 0 |xn 0| < 1M . 0 < xn M . 1xn +.

.[] xn 6= 0 n |xn| +. 1|xn| 0 , ,

1xn

0.

2.3.26. (xn) , xnn 0. , 1+(1)

n1

n 0. :n3[n/3]

n 0 sinnn 0.

2.3.27. ((1)n1n

)

(1)n1n = n +. 1

(1)n1n 0. ,1

(1)n1n =(1)n1

n 0. ((1)n1n

) .

x = 0 2.9[]; 10 .

2.3.28. (1)n1

n 0, ((1)n1n

)

.

2.3.28;

( (1)n1n

), ,

. (xn) , ( 1xn ) +. , (xn) , ( 1xn ) . , (xn) , . xn 0

40

|xn| 0 1|xn| +. 1|xn| 1, ,

, 1xn 1 n 1xn

1 n. ( 1xn ) .

, , : 10 0 0 .

0+ 0 0 0 , , 2.11[].

. 10+ = +,

10 = .

1.1 : , , , .

, 2.12, 4.3. , ( ) . - 1.4 ab. 00, 1+,1, (+)0 0.

. (xn) (yn) (xn

yn).

2.12. xn > 0 n. xn x R yn y R xy , xnyn xy. , xn 0 yn , xn

yn +.

2.3.29. 2.12, na 1 a > 0.

, (a) ( 1n). a a 1n 0,

na = a1/n a0 = 1.

2.3.30. (an) a > 1 0 < a < 1 ,, 2.12. (a) (n). a a n +, an a+, +, a > 1, 0, 0 < a < 1.

2.3.31. nn 1 2.12.

(n) ( 1n), n + 1n 0, (+)

0 .

0 , , , - 2.12 .

. (0+) = +.

.

2.3.1. ( (n+1)27(n+3)79

(2n+1)106

),(n3+(1)nn2+1

3n2+2(1)n1n),(n(n+1)

n+4 4n3

4n2+1

),

((1 n)5 + n4),(( n

3+n+13n2+3n+1

)9),( 3n+(2)n3n+1+2n+1

), (n+ 1

n), (

n2 + n+ 1

n2 + 1).

41

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+ (x+1)2n

(2x+1)n ;

2.3.6. x 6= 1 xn 6= 1 n. xn x xn

1xn x

1x .

2.3.7. 11 3+(1)n

2n 0, 3+(1)n

2n > 0 n (3+(1)n2n

) .

(3(1)n1)n

2 + ( (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 R yn y R. x < y, xn < yn.

2.3.10. xn x R yn y R x 6= y. |x y| > a, |xn yn| > a.

2.3.11. , xn [l, u] n xn x, x [l, u]. x (xn), xn (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 +, 22n+(1)n1n 0,

(12 +

(1)n14

)n 0.2.3.14.

[3n2n+1

n+2

] +, [

n]n

1 n+23n2n+1

[3n2n+1

n+2

] 1.

2.3.15. (1 + 1n)n2 +, (1 1

n2)n 1, (1 + 1

n2)n 1.

2.3.16. 12 [nx]

+, x > 00, x = 0, x < 0

[nx] [ny]

+, x > y0, x = y, x < y

nx [ny]

+, x > y 0, x = y Z , x < y , x = y Q \ Z

2.3.17. (2n)!(n!)2

+.

2.3.18. 0 a b c. nan + bn b, n

an + bn + cn c.

11 0 +, - , , .

12 x = y R \Q 2.7.18.

42

2.3.19. nn3 1, n

n4 + 3n2 + n+ 1 1.

2.3.20. a, [a]+[2a]++[na]n2

a2 .

2.3.21. nn2+1

+ nn2+2

+ + nn2+n

1 1n2+1

+ 1n2+2

+ + 1n2+n

1.

2.3.22. limm+(limn+(cosm!x)2n

)=

{1, x Q0, x R \Q

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, max{xn, yn} max{x, y} min{xn, yn} min{x, y}.

2.3.25. : n 1n =1n + +

1n (n ) 0 + + 0 (n ) = n0 = 0.

: (1 + 1n)n = (1 + 1n) (1 +

1n) (n ) 1 1 (n ) = 1

n = 1. 2.9[,];

2.3.26. x1 > 0 xn+1 x1 + + xn n N. 0 < a < 2, xnan +. a = 2 (2

n).

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 +, yn (xn + yn) (i) c R (ii) . (xn), (yn) xn 0, yn + (xnyn) (i) c R (ii) . (xn), (yn) xn 0, yn 0 (xnyn ) (i) c R (ii) . (xn), (yn) xn +, yn + (xnyn ) (i) c [0,+] (ii) . (xnyn ) c [, 0);

2.3.30. 14 (xn), (yn) xn +, yn 0 (xn

yn)(i)

c [0,+] (ii) . (xn

yn)

c [, 0); (xn), (yn) xn 0, yn 0

(xn

yn)(i) -

c [0,+] (ii) . (xn

yn) c [, 0);

(xn), (yn) xn 0, yn (xn

yn)(i)

c {+,} (ii) . (xn

yn) c R;

2.3.31. (xn), (yn) xn, yn > 0 n, xn 0, yn + (xnyn) .

13 .14 . 2.4.10.

43

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), {2}, [0, 1]{2} supremum . , ( ) , , . , , , . N, Z, Q, { 1n |n N}, (0, 1] (R \ Q), (0, 2) Q , supremum infimum .

2.3.34. - A u A. u = supA A u. infA l A.

2.3.35. - A. supA A A supA. supA / A, A supA. infA.

2.3.36. k N, k 2 xn 0 n. xn x, k

xn k

x.

xn +, kxn +.

2.12.

2.3.37. |xnxm| 1 n,m n 6= m. |xn| +. (xn) ; (n), (n), ((1)n1n).

2.3.38. xn x xn x n, sup{xn |n N} = x. x < y xn < y n xn x, sup{xn|n N} < y.

2.3.39. xn x, k inf{xn |n k} x sup{xn |n k}.

2.3.40. xn x. (xn) k xk x. (xn) k xk x.

2.3.41. 15 [] Cesro: xn x R, x1++xnn x. xn = (1)n1 n, x1++xnn 0. xn =

1+(1)n2 n n,

x1++xnn +.

(xn) . - Cesro.[] an+1 an a R, ann a.

15 2.7.13.

44

[] Cesro: 16 (xn), (yn) yn > 0 n y1 + + yn +. xnyn l R,

x1++xny1++yn l.

[] xn > 0 n xn x [0,+], nx1 xn x.

[] an > 0 n an+1an a [0,+], nan a.

( nn),

(nn!)

(n

(2n)!/(n!)2).

2.4 .

2.1 . , , , , . - : ((1)n1) ((1)n1n) .

2.1 .

2.1. 17 . :[] (xn) , limn+ xn = sup{xn |n N}.: (xn) , +, , . - .[] (xn) , limn+ xn = inf{xn |n N}.: (xn) , , , . - .

. [] - {xn |n N}. ( ) supremum , ( ) supremum +. (xn) .

sup{xn |n N} = +

xn +.M > 0. M {xn |n N}, n0 xn0 > M . (xn) ,

xn xn0 > M

n n0. xn +. (xn) .

x = sup{xn |n N}

xn x. > 0. x < x, x {xn |n N}. n0 x < xn0 . (xn) ,

x < xn0 xn16 Cesro yn = 1 n.17 supremum 2.4.17.

45

n n0. , x {xn |n N},

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 . , , ( ) .18

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. , - 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) 18, , 2.3.23.

46

, , ., , 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 +, , xn+1 =

3+(6/xn)1+(4/xn)

n, + = 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 +

11(1/2) = 3.

n. (xn) , , .

2.4.3. ((1 + 1n)

n) , ,

.19

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!(11n) (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!(11

n+1) (1n1n+1) +

1(n+1)!(1

1n+1) (1

n1n+1)(1

nn+1).

(2.3)

19 2.4.4.

47

(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 +, ( 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+(1 + 1n

)n= limn+

(1 + 11! +

12! + +

1n!

).

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+ ( 1.4 2.3). xn 1 yn + :xnyn 1. : xnyn 1yn = 1 1 . , , (1+ 1n)

n e . , 1 + 1n 1 n + (1 +

1n)

n 1.

. e - y > 0, loge y,

log y ln y.

48

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. 20

1 + 12 +13 + +

1n +.

xn = 1 + 12 +13 + +

1n n.

xn+1 xn = 1n+1 > 0 n, (xn) , , . n

x2n xn = 1n+1 + +1

n+n 1

n+n + +1

n+n = n1

n+n =12 .

(xn) x, , (xn) , xn x2n x x. xn x, x2n x. x2n xn x x = 0 , x2n xn 12 n. xn +. xn + . , 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) + 1

2k+1+ + 1n

1 + 12 + (14 +

14) + (

18 +

18 +

18 +

18) + + (

12k

+ + 12k)

= 1 + 12 + 214 + 4

18 + + 2

k1 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 +, xn +.

2.4.5. 21

(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) , , .20 2.5.4, 2.6.2 7.3.20

8.2.7, 8.2.10 8.3.1. 2.4.6, 6.4.11.21 2.6.1, 8.2.7 8.2.10 6.4.11, 7.3.20 8.2.1.

49

. [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 [a, b] an a x b bn n. , x an x bn n, a x b, x [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 [an, bn]. , an x bn n x [an, bn], . : [an, bn] , +

n=1[an, bn] = [a, b],

a = limn+ an b = limn+ bn.

2.4.6. p N, p 2. x [0, 1),

0 x < 1. (2.5)

, x p [0, 1p

),[1p ,

2p

), . . . ,

[p1p , 1

) 1p . x; kp x 1, (xn) . x1 < 1 x1 6= k1k k N, (xn) . x1 = k1k k N;[] x1, x2 > 0 xn+2 = xn+1 + 2xn n. (xn+1xn ) .[] xn+1 = sinxn n. (xn) , , .

2.4.9. 28 [] a > 1, (an) , an+1 = aan, an +. 0 < a < 1.[] a > 1 k N, (an

nk) , -

an+1

(n+1)k= a n

k

(n+1)kan

nk, a

n

nk +.

[] a > 1, ( na) , -

(

2na)2

= na, n

a 1. a = 1, 0 < a < 1;

[] ( nn)

nn 1.

[] (an

n! ) , an+1

(n+1)! =an

n!a

n+1 , an

n! 0.

2.4.10. 29 (xn), (yn) xn 1, yn + (xn

yn)(i)

c [0,+] (ii) . (xn

yn)

c [, 0); (xn), (yn) xn 1, yn

(xn

yn)(i)

c [0,+] (ii) . (xn

yn) c [, 0);

2.4.11. [] b , 1b+1+1

b+2+1

b+3+ +1

b+n +.[]30 (1 + a1) (1 + an) 1 + a1 + + an, a1, . . . , an 0, (1 a1) (1 an) 1 a1 an, 0 a1, . . . , an 1. b , limn+

(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+1

xn 1

) c, xn +.

27 2.5.5.28 2.3.19 2.3.22 2.3.25.29 . 2.3.30.30 (1 + a)n 1 + na Bernoulli a 0

1 a 0, .

53

c < 0 n(xn+1

xn 1

) c, xn 0.

c > 0 n(xn+1

xn 1

) c, xn +.

2.10. (2n)!

4n(n!)2 0 enn!nn +.

2.10;

2.4.12. 31 y 0 k 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+ xn, xk = y x 0.

2.4.13. xn+1 xn+xn+22 n.32

, , (xn) . (xn xn+1) xn xn+1 0. (xn) . (xn) . (xn) , , , .

2.4.14. 0 < a b. x1 = a y1 = b xn+1 =

xnyn yn+1 = xn+yn2 n,

(xn) , (yn) , xn yn n (xn), (yn) , GA (a, b).33

w1 = a z1 = b wn+1 = 2wnznwn+zn zn+1 =wnzn n,

(wn) , (zn) , wn zn n (wn), (zn) , HG (a, b).34

a, b,HG (a, b),GA (a, b) H (a, b) = 2aba+b , G (a, b) =

ab A (a, b) = a+b2 ;

2.4.15. xn = 135(2n1)246(2n) n. (nxn2)

((n+ 12)xn

2) .

.

2.4.16. [] f : [a, b] R [a, b]. f(a) > a f(b) < b, (a, b) f() = .[] I f : I R x I > 0 f(x) f(x) f(x) x, x (x , x + ) I x < x < x. f I .

2.4.17. . - , supremum.35

. 12n 0., - A. x1 A y1 A. [x1, y1] A A. x1+y12 A, x2 = x1, y2 =

x1+y12 , ,

x2 =x1+y1

2 , y2 = y1. [x2, y2] A 31 1.2.32 (xn) . n, -

.33 GA (a, b) - a, b.34 HG (a, b) - a, b.35 supremum : .

54

A. , [x1, y1], [x2, y2], . . . [xn+1, yn+1] [xn, yn] yn xn = y1x12n1 n [xn, yn] an A un A. u xn u, yn u , , an u, un u. u A.

2.4.18. 36 I , . ( ) (an) I = {an |n N}. [x1, y1] I y1 x1 > 0 a1 / [x1, y1]. [x2, y2] [x1, y1] y2 x2 > 0 a2 / [x2, y2]. , [x1, y1], [x2, y2], . . . [xn+1, yn+1] [xn, yn] an /[xn, yn] n. [xn, yn] , , 6= an n. I .

2.4.19. 37 1 , n 2, 2n 2n . pn qn , , . p2 = 4

2 q2 = 8

pn+1 = 2pn(2 +

(4 pn

2

4n

)1/2)1/2 qn+1 = 4qn(2 + (4 + qn24n )1/2)1 n 2. qn = pn

(1 pn

2

4n+1

)1/2 n 2. (pn) , (qn) pn < qn n 2. (pn), (qn) .38

2.4.20. 39 x (rn) rn x. , p- x - 2.3.32. . y > 1 x . (rn) rn x (yrn) . - (rn) rn x (yrn) . yx = limn+ yrn (rn) rn x. yx = limn+ yrn ( ) (rn) rn x. y > 1 x . yx = limn+ yrn ( ) (rn) rn x. 0 y 1 x . yx - ( 1.4). 1.8 .

2.4.21. 40 y > 0.

(n( n

y 1)

) .

. log y = limn+ n( n

y 1).

36 A, R, a : N A A. an = a(n), A A = {an |n N} , , A . A , .

37 .38

, , , 2, . , pn 2 qn n 2. pn 2 qn 2.

39 .40 .

55

log y = limn+ yxn1xn

(xn) xn 0 xn 6= 0 n. 2.13 ( []) . a > 0, a 6= 1 y > 0. loga y =

log ylog a 1.9.

2.5 .

. (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, . . . ).

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 N nk < nk+1 k. nk k k.

56

. n1 1 n1 N. nk k k. nk+1 > nk nk, nk+1 N, nk+1 nk + 1 , , nk+1 k + 1. nk k k.

2.2

nk +.

2.14. , .

. xn x R (xnk) (xn). xnk x. > 0. n0 xn Nx() n n0. nk +. k nk n0 , , xnk Nx(). xnk x.

2.14 , , : , .41

2.5.6. ((1)n1) ., (1)(2k1)1 = 1 1 (1)(2k)1 = 1 1.

2.15 .

2.15. x R x2k x x2k1 x. xn x.

. > 0. x2k x, xn Nx() n n . , x2k1 x, xn Nx() n n . xn Nx() n ( n n) . xn x.

2.5.7. 42

(1 12 +

13

14 + + (1)

n1 1n

).

xn = 1 12 +13

14 + + (1)

n1 1n n.

x2k+2 x2k = 12k+1 1

2k+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 < . . . < x2n < x2n+2 < . . . < x < . . . < x2n+1 < x2n1 < . . . < x3 < x1.

41 . 2.5.15.42 2.6.3 6.4.11 8.3.9.

57

(x2n) x , , (x2n1) x. , , [x2, x1],[x4, x3], [x6, x5], . . . x - .

. . , ((1)n1) . , ((1)n1), , : 1 1. ((1)n1) . , .

BOLZANO - WEIERSTRASS. .

. 43 (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 [l1, u1]. [l1, u1] [l1, l1+u12 ], [

l1+u12 , u1]. [l1, u1] -

(xn), (xn). [l2, u2] ( ). [l2, u2] [l1, u1], u2 l2 = u1l12 [l2, u2] (xn). (xn) ( ) [l2, u2]: xn2 [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 [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 [lk, uk]

k. uk+1 lk+1 = uklk2 k uk lk =

ul2k

k,

uk lk 0.43 2.5.9.

58

, (lk), (uk) . lk x uk x. nk+1 > nk k, (xnk) - (xn) , lk xnk uk k, xnk x.

+ . - . (1, 0, 3, 0, 5, 0, 7, . . . ) +., , +, +: , (1, 3, 5, 7, . . . ). .

2.16. [] - +.[] .

. 44 [] (xn) . u (xn) > u. 45

( ) u (xn) > u. (xn) n0 (, u]. (xn) (, u] x1, . . . , xn01. , , , (, u] (, u] (xn). (xn) . . (xn) +, . (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 +.[] .

.

2.5.1. a 0. ex, r, s Q r < x < s ex < er < es < ex + . , ex < (1 + rn)

n < (1 + xn)n < (1 + sn)

n < ex + (1 + xn)n ex.

44 2.5.9.45 2.4[]

2.4[].

59

2.5.4. (xn) (xnk) xnk x R, xn x. (xn) (xnk) xnk x R, xn x.46 xn = 1 + 12 +

13 + +

1n n. 2.4.4

x2n xn 12 n. x2k k2 + 1 k ,

, xn +.

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).

2.5.7. a, b, x 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 R x3k x x3k1 x x3k2 x. - 2.15, xn x. . a, b, c, x 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 N n = 2m1k m N k N. N : A(m) = {2m1k | k } m N. A(1) , A(2) - , A(3) . , xn = 1k n = 2

m1k m N k N. (xn) : (x

(m)k )

+k=1 m N, ,

m N, x(m)k =1k k N. ,

0. (xn) 0, (x2m13)+m=1.

2.5.9. (xn). xn - m > n xm > xn.47

46 2.4.4. 2.6.2 7.3.20 - 8.2.7, 8.2.10 8.3.1 2.4.6 6.4.11.

47 xn xm +.

60

(xn) -, (xn). (xn) -, - , , . . Bolzano - Weierstrass 2.16.

2.5.10. (xn) n (xnk) k.

2.5.11. 48 , - .

2.5.12. (xn) x R (xn) x. : (xn) (xn) .

2.5.13. xn < x n. sup{xn |n N} = x (xn) x. ( 1n). sup{

1n |n N} = 1. , (

1n)

0, ( 1n), 1. - ;

2.5.14. (xn) (xnk). (xnk) (xn).

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 Z qn N n. qn +.[] x (rn) rn x rn = pnqn pn Z qn N n. qn + pn x(+).

2.6 .

. (xn) Cauchy > 0 n0 |xn xm| < n,m n0. :

limn,m+(xn xm) = 0.

: (xn) Cauchy .

2.17. (xn) , Cauchy.48 2.1.9.

61

. xn x. > 0. n0 |xn x| < 2 n n0. , n m, |xm x| < 2 m n0.

|xn xm| = |(xn x) (xm x)| |xn x|+ |xm x| < 2 +2 =

n,m n0. (xn) Cauchy.

Cauchy 2.17.

CAUCHY. (xn) Cauchy, .

. 49 (xn) Cauchy. n0 |xn xm| < 1 n,m n0. (m = n0), n n0 |xn xn0 | < 1,

|xn| = |(xn xn0) + xn0 | |xn xn0 |+ |xn0 | < 1 + |xn0 |.

, M = 1+ |xn0 |, (xn) [M,M ], (xn) . (xn) , , Bolzano - Weierstrass, - (xnk) .

xnk x.

xn x. > 0. n0

|xn xm| < 2 n,m n0.

n n0 , , . nk +, nk n0 , ,

|xn xnk | < 2 . (2.9)

, xnk x,

|xnk x| < 2 . (2.10)

(2.9) (2.10). k (2.9) (2.10) k

|xn x| = |(xn xnk) + (xnk x)| |xn xnk |+ |xnk x| < 2 +2 = .

|xn x| < n n0, xn x.

Cauchy 2.1. (xn) x (xn), |xn x| x, |xnxm| . , , Cauchy Cauchy.

49 2.6.6, 2.6.7 2.7.14.

62

2.6.1. (xn) xn = 1+ 122 +132

+ + 1n2

n. , ( 2.4.5.50) (xn) .m > n.

|xn xm| = 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 1

n+1) + (1

n+1 1

n+2) + + (1

m2 1

m1) + (1

m1 1m) =

1n

1m 0. n0 1n0 < . m > n n0 |xn xm| < 1n

1n0< .

(xn) Cauchy , , .

Cauchy R.

. Cauchy .

Bolzano - Weierstrass, - , , supremum.

.

2.6.1. (xn), (yn) Cauchy. , , (xn+yn),(xnyn) Cauchy.

2.6.2. 51 xn = 1 + 12 + +1n n, 2.4.4

x2n xn 12 n. (xn) Cauchy; xn +.

2.6.3. 52 xn = 1 12 +13

14 + +

(1)n1n n.

|xn xm| = 1n+1

1n+2 + +

(1)mn1m

1n+1 n,m n < m. (xn) Cauchy .

2.6.4. [] 0 M < 1 |xnxn+1| cMn. n0 |xn xm| c M

n

1M n,m n0 n < m. (xn) Cauchy. x (xn), |xn x| c M

n

1M .[] 0 M < 1 |xn+1 xn+2| M |xn xn+1|.53 (xn) Cauchy. x (xn), c 0 |xn x| c M

n

1M .[] x1 > 0 xn+1 = 1 + 31+xn n. (xn) . n- xn .[] || < 1 xn+1 = a+ sinxn n. (xn) . n- xn .

2.6.5. . , - , supremum.54

50 8.2.7 8.2.10 6.4.11, 7.3.20 8.2.1.51 2.4.4. 2.5.4 7.3.20

8.2.7, 8.2.10 8.3.1 2.4.6 6.4.11.52 2.5.7. 6.4.11 8.3.9.53 (xn) . 0 M 1, .54 supremum -

.

63

2.4.17 . , -. 12n 0 , , (xn), (yn) Cauchy. - .

2.6.6. 55 Cauchy (xn). [a1, b1] b1 a1 < 1 xn [a1, b1]. - [a2, b2] [a1, b1] b2 a2 < 12 xn [a2, b2]. , [ak, bk] k [ak+1, bk+1] [ak, bk] bk ak < 1k k xn [ak, bk] k. x x [ak, bk] k , , xn x.

2.6.7. 56 Cauchy (xn). ln = inf{xk | k n} un = sup{xk | k n} 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 R. x (xn) (xn) x.

2.7.1. (xn) , , . , ( 1n) 0 (n) .

2.7.2. (xn) xn = (1)n1 n - , 1 1. (x2k) 1 (x2k1) 1. x (xn). (xnk) (xn) x. (x2k) (x2k1) (xn), (xnk) (x2k) (x2k1). (xnk) (x2k), (xnk) (x2k) - , (xnk), x , (x2k), 1 , , x = 1., (xnk) (x2k1), x = 1., x (xn), x = 1 x = 1. (xn) 1 1.

2.7.3. , ( (1)n1+1

2 n)

0 +.

Bolzano - Weierstrass 2.16, : , - , , . - -.

55 Cauchy.56 Cauchy.

64

2.2. - .

. : (xn) . + (xn) , , - (xn). : (xn) . , , (xn). : (xn) 6= . (xn) , u xn u n. (xn) , , (xn) u. (xn) - 6= u. L (xn),

L = {x R |x (xn) }.

- R. L supremum -

x = supL.

x (xn). x L, L , , (xn). > 0. x = supL, x L x < x x , ,x < x < x + . x L, (xn) x. (x, x+) , , (xn) (x , x+ ). = 1, 12 ,

13 , . . . .

(xn) (x 1, x+1), , xn1 ,

x 1 < xn1 < x+ 1.

(xn) (x 12 , x+12), ,

xn2 , x 12 < xn2 < x+

12 n2 > n1.

(xn) (x 13 , x+13), ,

xn3 , x 13 < xn3 < x+

13 n3 > n2.

, (xnk) (xn)

x 1k < xnk < x+1k k

, ,xnk x.

x , , (xn). , (xn) . (xn) .

65

. 57 (xn) (xn)

lim supn+

xn lim supxn limxn.

(xn) (xn)

lim infn+

xn lim infxn limxn.

2.18. [] limxn :(i) limxn < x, xn < x.(ii) x < limxn, x < xn n.[] limxn :(i) x < limxn, x < xn.(ii) limxn < x, xn < x n.

. [] (i) limxn < x xn < x. (xn) x. (xn) x. , , (xn) x . x, (xn) x. , limxn < x limxn (xn).(ii) x < limxn. limxn (xn), - (xn) limxn. > x, (xn) > x.[] .

. , , , .

2.19. (xn).[] limxn limxn.[] (xn) limxn = limxn. , (xn) , limxn = limxn = limxn.

. [] , limxn limxn (xn).[] (xn) . limxn , , (xn). limxn = limxn = limxn. ( 2.7.1.), limxn = limxn. limxn = limxn = +, (i) 2.18[] x x < xn , , limxn = +. limxn = limxn = , (i) 2.18[] x xn < x , , limxn = ., limxn = limxn = x R. (i) 2.18[] > 0 xn < x+ (i) 2.18[] > 0 x < xn. > 0 x < xn < x+ . limxn = x.

57 2.7.15 .

66

.

2.7.1. a < b < c (a, b, a, b, a, b, a, b, . . . ) (a, b, c, a, b, c, a, b, c, . . . ). lim lim .

2.7.2. , , lim, lim : (n+1n ), ((2)n),

(2(1)

n1n),(

(1)n1(1 1n)),(2n3[n/3]

).

2.7.3. limxn = + (xn) . limxn = xn . limxn ;

2.7.4. limxn = lim(xn).

2.7.5. xn yn, limxn lim yn limxn lim yn.

2.7.6. an bn cn. lim cn lim an, (an), (bn), (cn) .

2.7.7. limxn + lim yn lim(xn + yn) lim(xn + yn) limxn + lim yn . limxn lim yn lim(xnyn) lim(xnyn) limxn lim yn xn > 0 yn > 0 . t > 0, lim(txn) = t limxn lim(txn) = t limxn. t < 0;

2.7.8. (yn) , lim(xn + yn) = limxn + lim yn lim(xn + yn) = limxn + lim yn. xn, yn > 0, (yn) , lim(xnyn) = limxn lim yn lim(xnyn) = limxn lim yn.

2.7.9. m N, (xn), (xn+m) .

2.7.10. (xnk) (xn). 2.5.14 (xnk) - (xn). limxn limxnk limxnk limxn.

2.7.11. [] x R (xn) > 0 xn Nx() n.[] X R (xn). (yn) X yn y R, y X . a 0 n, limxn lim n

x1 xn lim n

x1 xn

limxn.[] yn > 0 n y1+ +yn +, lim xnyn lim

x1++xny1++yn

lim x1++xny1++yn limxnyn.

67

2.7.14. 58 Cauchy (xn). x = limxn R x = limxn R. Cauchy (xn) , x, x R. x < x. l, u x < l u

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