mathimata seirwn

42
Κωνσταντίνος Σπ. Κατωπόδης Δρ Μαθηματικός Καθηγητής ΤΕΙ Θεσσαλονίκης ΑΚΟΛΟΥΘΙΕΣ - ΣΕΙΡΕΣ ΔΙΩΝΥΜΙΚΕΣ ΔΥΝΑΜΟΣΕΙΡΕΣ Περιληπτική Θεωρία και Ασκήσεις Θεσσαλονίκη 2008

description

ΑΚΟΛΟΥΘΙΕΣ - ΣΕΙΡΕΣΔΙΩΝΥΜΙΚΕΣ ΔΥΝΑΜΟΣΕΙΡΕΣΠεριληπτική Θεωρία και ΑσκήσειςΘεσσαλονίκη

Transcript of mathimata seirwn

  • .

    -

    2008

  • i

    iii 1 1 1.1 1 1.2 - 3 1.3 5 1.3.1 5 1.3.2 6 10 2 11 2.1 11 2.1.1 13 2.1.2 19 2.2 21 2.2.1 25 27 3 29 3.1 29 3.2 30 3.3 31 34

  • . . : --

    ii

    35

  • iii

    , , . . , , . , . , . , . , . . . -, . , , . . . , , ... , . . ,

  • . . : -- iv

    . . . . - -. , , . . ' .

    , 2008

    /

  • 1

    1.1 1.1:

    :na N R (1.1) (sequence) 1. n=n(n) (n)nN (n).

    1, 2, 3, , n, . (1.2) (terms) . n. n. 2 .

    1 ={1, 2, 3, , n,}, ( , )R = , { , }R R= . 2 .

  • . . : -- 2

    1.1: (n) n=1n

    , n= 2 !

    n

    n

    nn

    n .

    1 2 31 1 1 11, , ,..., , ...1 2 3 n

    a a a an

    = = = = =

    1 2 3

    1 2 31 2 3

    1 2 3, , ,..., , ...2 1! 2 2! 2 3! 2 !

    n

    n n

    na a a an

    = = = =

    1 2 31 1 9, , , ..., , ...2 2 16 2 !

    n

    n n

    na a a an

    = = = = .

    1n n na a a 2 = + , 1=1, 2=2

    1=1, 2=2, 3=3, 4=5, 5=8, 6=13, 7=21, .

    ..

    12, 22, 32, (n) n=n2

    12, 22, 32, 42, , n2, , (n)

    n=n35n2+11n6

    12, 22, 32, 22, 49, 96, , , n35n2+11n6, , 12, 22, 32. .... ..

    1 1 1 1 1 1 1, , , , , , ,..2 3 5 7 11 13 17

    . .

    . , , . 1.1: , .. . 1.2 -

  • . 1: 3

    (increasing sequence) (decreasing sequence). 1.2: ()

    nN, nn+1 (nn+1). (1.3)

    nN, nn+1) (1.4)

    ( )1. 1.3: M ( ) (m)

    nN, n (nm). (1.5)

    m - . 1.4: m

    nN, m n . (1.6) m - . m m (upper bound) (lower bound) . 1.1: =max{|M|, |m|}

    nN, |n|. (1.7)

    . 1.2: :

    () n=(1)nn, () n=2+ 5n

    , () n=4

    2n

    n5+ , () n= 1

    2 3n

    +

    (i) (ii) (iii) . : () (i) 1, 2, 3, 4, 5, 6, ,(1)nn, . (ii)

    1na a+ n =(1)n+1(2n+1) 1 nN , n0: n n0 (. 1.1).

  • . . : -- 4

    . (iii) |n|=n |n|M. .

    () (i) 7, 92

    , 134

    , 3, , 2+ 5n

    ,

    (ii)

    1n na a+ =2+ 5 1n + 52n

    + = 5

    ( 1n n )+ 2 n=2+5n

  • . 1: 5

    12

    32

    .

    1.3 - 1.3.1 () R 1.4: (n) (R) 1

    niml n=

    >0, , n0=n0()2: nn0 na

  • . . : -- 6

    (n) (n) niml n=1

    niml n=2.

    .

    () (kniml n)=k1, kR.

    () (niml n n)=1 2.

    () (niml n n)=1 2.

    () niml n

    n

    =

    1

    2

    , 20 n0, nN.

    () = ( =n iml ( )kna )knn im al 1k , kR. 1.3.2 (divergent sequence). : 1.6: () (n) +,

    niml n=+, M>0 n0=n0(M): n n0

    n>M.

    () (n) ,

    niml n=, M>0 n0=n0(M): n n0 n0, , n0=n0(): n n0 na

  • . 1: 7

    ()

    1n + n < , 1n + n 1

    1n + > 12

    n> 21 1

    4 n0= 21 14

    .

    1.4: :

    () n= 2 1n

    n , () n= 1n + n , () n=4

    2n

    n5+ , () n= ( )n nn

    l .

    : () niml 2 1

    nn = n iml

    2

    2

    2 21

    nn

    nn n

    =

    niml

    2

    1

    11

    n

    n

    =()= 01 0 =0.

    1: niml

    4

    2

    3 25 6n nn n ++

    71

    =. (;) () ( )

    niml ( 1n n+ )=

    niml 1

    1n n+ + = n iml1

    11 1

    n

    n+ +

    =niml

    1

    11 1

    n

    n+ +

    =0.

    () niml 4 5

    2n

    n+ =(())= 4

    2=2.

    () ( )n n n0, () n= n n , () n=n, >0, () n= 1nk

    n +

    , kR. : () (n) .

    (i) >1 n a >1 n a =1+n =(1+n)n>1+nn>nn 0

  • . . : -- 8

    niml n a = (1+

    niml n)

    niml n a =1+

    niml n

    niml n a =1.

    (ii) 2( 1)2 n

    n n

    + 0< 2n < 2 1n +

    0< n 1+n>n. n n,

    niml n = .

    (ii)

  • . 1: 9

    : niml

    1

    nnn

    = n iml1

    1

    n

    nn

    =

    niml 1

    11n

    n +

    = 111

    n

    nim

    n + l

    = 11

    e=e.

    3: n=3

    3 1

    nnn

    + . :

    33 1

    nnn

    + = n iml1

    3 13

    nnn+

    =niml 1

    113

    n

    n +

    = 13 3

    1

    113

    n

    nim

    n +

    l=

    niml

    = 13 3

    3

    1

    113

    n

    nim

    n + l

    = 13

    1

    e.

    4: n=2 1

    3 1

    nnn

    . :

    n=2 1

    3 1

    nnn

    =2 1

    3 1 3 1

    nn nn n

    =2 3 1

    3 1

    nn nn n

    = 21 3

    3 1 nnnn

    n

    1 =

    = 21 313

    nnn

    n

    1 = 22

    1 3 1

    13 13

    nn

    nn

    n

    = 21 3 1

    139 1

    n

    n

    nn

    n

    +

    .

    niml

    2 1

    3 1

    nnn

    = n iml 21 3

    139 1

    n

    n

    nn

    n

    1

    +

    =

    = 21 3 1

    13[ (9 )] 1

    n n

    n

    n n

    nimn

    im imn

    +

    l

    l l

    (9 )nniml=0, =,

  • . . : -- 10

    2131

    n

    nim

    n

    + l = ( )213e = 23e 3n nim n 1l =3.

    1. () (i)n= 2( 1)

    ( 1)

    n

    n+

    (ii) n= 3 2 1n

    n n+ + .

    () n=2

    100n +.

    n0 n>104.

    (.: () n0>1 1

    , () n0=103).

    2. :

    () n=2

    2

    23 1

    n nn n

    1 ++ , () n=

    ( 3) 23

    n n

    n

    + , () n= !3nn , () n=

    213 1nn+ .

    (.: () 23

    , () , () , () 19

    ).

    3. :

    () n: 1 2a = , 2 2 2a = + , , 1 2na + na= + , () n= 23n

    n

    () n=3

    2

    ( 2)1 1

    n n nn n+ + + , () n=

    11

    n

    n

    aa

    + , >0.

    (.: () 2, () , () 1, () ).

    4. :

    () n=!n

    nn

    , () n=2 2 21 1 ... 1

    2 3 3 4 ( 1)n n + .

    (.: () 0, () 13

    ).

  • 2

    2.1 2.1: (n)

    1 + 2 + 3 + + n +, (2.1)

    1n

    na

    = , (2.2)

    (numerical series). (converges) (diverges). (2.1) . , , (2.1) . 1. (Sn),

    S1 = 1, S2 = 1 + 2, , Sn = 1 + 2 + 3 + + n,

    (sequence of partial sums).

    1 .

  • . . : -- 12

    (2.1) , , . =s, sR, ( .. = ) .

    nnim Sl nn im Sl

    nnim Sl

    2.1: (n) n = n1, 0,

    1

    1

    n

    na

    = = + + 2 + 2 + + n1 (2.3)

    (Sn)

    Sn = + + 2 + 2 + + n1 (2.4)

    1. (2.4)

    (1 )1

    n

    naS

    = . (2.5)

    (2.5) : () ||1 ( ).

    11

    nn

    a s

    == , 2

    1n

    ns

    == , 1 2,s s R .

    () k .

    11 1

    n nn n

    ka k a ks

    = == = . (2.7)

    ()

    1 1

    ( )n n nn n

    a a1

    nn

    = =

    = = , (2.8)

    , , . () - .

    () 1

    nn

    a

    = im 0nn a =l . (

    ).

    1 (n) . .

  • . 2: 13

    2.1: 0nn im a = l . (;) 2.2: , ,

    1

    32nn

    = . (2.9)

    : 32

    12

    .

    12

    1 1 . 2

    1

    1n n

    = =2>1, 3

    1

    1n n

    =

    1

    1n n

    =

    = 13

  • . . : -- 14

    .

    1 1

    | |n nn n

    a a

    = == .

    2.1.1.1 , ,

    . 1

    nn

    a

    =

    1n

    n

    =

    .

    1. : () nn 1

    nn

    =

    . 1

    nn

    a

    =

    () nn 1

    nn

    a

    =

    1n

    n

    = .

    2.4: :

    () 2

    1

    11n n

    = + , () 311

    n

    nn

    =

    + , () 1

    12nn n

    = , () 12 1

    2

    n

    n

    nn

    =

    + . : 4 . ()

    2

    11n + 2 2

    1n n+ =

    1 12 n

    1

    1n n

    = ,

    1

    1 12n n

    = (. ()),

    1() . ()

    31

    1n

    nn

    =

    + = 2 31

    1 1n n n

    =

    + = 211

    n n

    = + 3

    1

    1n n

    = .

    21

    1n n

    = 3

    1

    1n n

    = 2 3-

    , () . ()

    12nn

    12n

  • . 2: 15

    1

    12nn

    = , , 1()

    . ()

    niml 2 1

    2

    nnn+ = n iml

    112

    n

    n + = n iml

    12 211

    2

    n

    n +

    =1

    2e = e 0

    1.1 .

    2. D Alembert: 1nn

    n

    aima+

    l =k. k1 , k=1 1

    nn

    a

    =

    1. 2.5: :

    () 1

    12nn n

    = , () 2312

    3

    n

    nn n

    = () 1 2 !n

    nn

    nn

    = () 211

    4n n

    = + . : 4 . ()

    1n

    nn

    aima+

    l = im

    nl1

    1( 1) 2

    12

    n

    n

    n

    n

    ++

    =

    niml 1

    2( 1) 2

    n

    n

    nn +

    + = n iml

    12 1

    nn + =

    12 n

    iml

    1n

    n + =12

  • . . : -- 16

    = 12 n

    iml 11

    n

    n + =

    12

    e>1,

    D Alembert . ()

    1n

    nn

    aima+

    l =

    niml

    2

    2

    1( 1) 4

    14

    n

    n

    + +

    +=

    niml

    2

    2

    4( 1) 4

    nn

    ++ + = n iml

    2

    2

    4( 1) 4

    nn

    ++ + =

    =2

    2

    4( 1) 4n

    nimn

    ++ +l = 1 =1

    , , .. .

    2 2 2

    1 1 124 nn n n

    > =+ +1 .

    1

    1n n

    = ,

    1

    1 12n n

    =

    (. ()), .

    3. Cauchy: niml n na =k. k1 , k=1 1. 2.6: :

    () 1

    5 32 7

    n

    n

    nn

    =

    + , () 2

    1

    45

    n

    nn

    =

    () 2

    1 1

    n

    n

    nn

    =

    , () 31 nnn

    e

    = .

    : 4 . ()

    niml n na = n iml

    5 32 7

    n

    nnn+ = n iml

    5 32 7

    nn+ =

    52

    >1

    Cauchy . ()

    niml n na = n iml

    245

    n

    n n = n iml24

    5n n

    = 16

    25

    niml n n = 16

    251= 16

    251 .

  • . 2: 17

    Cauchy . ()

    niml

    2

    1

    n

    nn

    n = n iml 1

    nnn

    = n iml111

    n

    n

    = 11

    e=e>1

    Cauchy . ()

    niml 3n n

    ne

    =niml 3

    n ne

    = 31e

    0. (;). -

    1

    nn

    = , 4n n = >0.

    niml

    2

    34 53 4

    4

    n nn n

    n

    ++ =

    niml

    3 2

    3

    4 512 16n n n

    n n ++ =

    412

    = 13R{0}

    1

    4n n

    = , ,

    2

    31

    4 53 4nn nn n

    =

    ++ .

  • . . : -- 18

    () 1n 0,

    2 (;), nN,

    1n

    >0.

    1

    1n n

    = .

    1

    1n n

    = .

    niml

    1

    1n

    n

    =1R{0}

    1

    1n n

    = 12 -,

    1

    1n n

    = .

    () 31

    43 4n n n

    = 5+ + . 2

    1

    1n n

    = ,

    2-.

    niml

    3

    2

    43 4 5

    1n n

    n

    +niml=

    2

    3

    43 4

    nn n

    +5+ + =0

    21

    1n n

    = ,

    31

    43 4n n n

    = 5+ + .

    () 2

    21

    4 54 5nn n

    n n

    =

    + + .

    1

    1n n

    = .

    niml

    2

    24 5

    4 51

    n nn n

    n

    + + =

    niml

    3 2

    2

    4 54 5

    n nn n

    n + + =

  • . 2: 19

    2

    21

    4 54 5nn n

    n n

    =

    + + .

    2.4: 2

    21

    4 54 5nn n

    n n

    =

    + +

    (. 4)

    niml

    2

    2

    4 54 5n n

    n n + + =

    140.

    2

    31

    4 53 4nn nn n

    =

    ++ 31

    43 4n n n

    = + + 5 . (;). 2.1.2 2.3: (plus and minus series), , . (2.1)

    1n

    na

    = =1+2+3++n+ . (2.10)

    (alternating series), , -

    1

    1

    ( 1)n nn

    a +=

    =12+3 +(1)n+1n +, (2.11) n0, nN1. , , . .

    : .

    1n

    na

    =

    2 1| n

    na

    =| 3

    . ( ).

    |

    1 n0 nn0, n0. 2 = |

    1| n

    na

    = 1| + |2| + |3| + + |n| +

    3 (absolutely convergent series).

  • . . : -- 20

    1.

    Leibnitz: .

    (

    1

    1

    ( 1)n nn

    a +=

    n) .

    2.8: :

    () 3

    1

    ( )1n

    nn

    = + , () 11

    1( 1)nn n

    +=

    , () 1 21

    1( 1)nn n

    +=

    () 11

    ( 1( 1)n nn

    nne

    +=

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

    3

    ( )1

    nn

    + 3

    ( )

    1

    n

    n

    +

    3

    11 n+ < 32

    1

    n.

    321

    1n n

    = , 32 -,

    3

    1

    ( )1n

    nn

    = + , ,

    , . () (i)

    1 1( 1)nn

    + = 1n

    .

    1

    1n n

    =

    .

    (ii) 1n

    , 1 1

    1n n> + ,

    , 1imn nl =0, , Leibnitz,

    2. () (i)

    12

    1( 1)nn

    + = 21n .

    1 , . 2 , (), .

  • . 2: 21

    21

    1n n

    = , 2-,

    .

    (ii) 21n

    , 2 21 1( 1)n n> + ,

    , 21im

    n nl =0, Leibnitz

    . ()

    1 1( 1)n nnne

    + + = 1nnne+ >0.

    niml

    1( 1) 1( 1)

    1

    n

    n

    nn e

    nne

    ++ +++ = n iml 2 1

    ( 2)( 1)

    n

    n

    n n en e +

    ++ = n iml 2

    1 ( 2)( 1)n n

    e n + + =

    1e1= 1

    e

  • . . : -- 22

    () x=x0

    nn

    na x

    = 0 0 x

    |x|>|x0|.

    x=0 1 = 0

    nn

    na x

    = 0 .

    . , , . (2.1) (2.12). 2.2: . , 2, R =0 =, (interval of convergence), (radius of convergence) . . (n),

    x ,

    .

    0

    nn

    na x

    =

    0

    nn

    na x

    = .

    D Alembert.

    niml

    11

    nn

    nn

    a xa x

    ++ =

    niml 1n

    n

    a xa+ = k x ,

    1nn

    n

    ak ima+

    = l . D Alembert

    0

    nn

    na x

    = k x

  • . 2: 23

    (, ) x= x=.

    0

    nn

    na x

    =

    2.1: -

    Caushy

    0

    nn

    na x

    =

    = 1n

    nnim al

    . (2.15)

    2.9: :

    () 3

    1

    2n nn

    xn

    = ()

    1

    1!

    n

    nx

    n

    = () 2

    1

    1 ( 3)nn

    xn

    = , ()

    1

    ( 1) ( 1)3 2

    nn

    nx

    n

    =

    + .

    : () ( (2.14))

    =1

    n

    nn

    aima +

    l =niml

    3

    1

    3

    2

    21

    n

    nn

    n

    +

    +=

    niml

    3

    1 3

    2 12

    n

    n

    nn++ =

    niml 31 1

    2n

    n+ = 1

    2 niml 3 1n

    n+ = 1

    2.

    ( 12

    , 12

    ).

    .

    (i) = 12

    3

    1

    2 1( 1)2

    nn

    nn n

    = 1

    31

    1( 1)nn n

    =

    1 (;).

    (ii) = 12

    3

    1

    2 12

    n

    nn n

    = 1

    31

    1n n

    =

    (;).

    1 1,2 2

    .

    ()

    =1

    n

    nn

    aima +

    l =niml

    1!

    1( 1)!

    n

    n +=

    niml ( 1)

    !n

    n!+ = (n+1)=

    niml

    R .

    1 .

  • . . : -- 24

    () t=x3 21

    1 nn

    tn

    = .

    =1

    n

    nn

    aima +

    l =niml

    2

    2

    1

    1( 1)

    n

    n +=

    niml

    2

    2

    ( 1)nn+ =1.

    (1,1). .

    (i) =1 21

    1( 1)nn n

    = .

    (ii) =1 21

    1n n

    =

    .

    21

    1 nn

    tn

    = 1 t 1 1 x3 1

    2 x 4, [2, 4].

    () t=x1 1

    ( 1)3 2

    nn

    nt

    n

    =

    + .

    =1

    n

    nn

    aima +

    l =niml

    13 2

    13( 1) 2

    n

    n

    ++ +

    =niml 3 5

    3 2nn++ =1.

    (1,1). .

    (i) =1 1

    13 2n n

    = + (;). (ii) =1

    1

    ( 1)3 2

    n

    n n

    =

    + (;).

    1

    ( 1)3 2

    nn

    nt

    n

    =

    + 1< t 1

    1< x1 1 0< x 2 (0, 2]. 2.2.1 -

    1

    n

    nx

    = =1 + x + x2 + x3 + + xn + (2.16)

    (1, 1), |x|

  • . 2: 25

    11 x .

    1

    n

    nx

    = = 1 + x + x2 + x3 + + xn += 11 x . (2.17)

    , , , x, x, , , x. (series of function) .

    1

    ( )nn

    f x

    = = f1(x) + f2(x) + f3(x) ++ fn(x) + . (2.18)

    x . x (domain of convergence) . - , (2.18), f(x). (, ) x s(x). . .

    () 0

    ( 1)n nn

    x

    = = 11 x+ () 1

    1 nn

    xn

    = = 1(1 ) 1n x n x = l l

    (2.19)

    () 10

    12

    nn

    nx

    +

    = = 12 x () 20 ( 1) nn n x

    =+ = 31(1 )xx+

    , . . , , 1. . (2.12) - (, ) s(x),

    0

    nn

    na x

    = = 0 + 1 x + 2 x2 + 3 x3 + + n xn + = s(x). (2.20)

    1 + 22 x + 33 x2 + + nn xn1 + = 11

    nn

    nna x

    = (2.21)

    (2.20). 1 .

  • . . : -- 26

    2.3: (2.20) (, ), (2.21) s(x).

    1

    1

    nn

    nna x

    = = 1 + 22 x + 33 x2 + + nn xn1 + (n+1)n+1 xn + = s(x).

    (2.3) . .. (, )

    22 + 63 x + + n(n1)n xn1 + = s(x)

    ... n(n1) (n2) (n3) (nr+1)n xn k + = s(k) (x), rN

    ( 1) ( 2) ... ( 1) n knn k

    n n n n r a x =

    + = s(k) (x), kN. (2.22) (2.20).

    2 3 1

    0 1 3 ... ...2 3 1

    n

    nx x xa x a a a

    n

    ++ + + + ++ =

    1

    0 1

    n

    nn

    xan

    +

    = + (2.23) (2.20). 2.4: (2.20) (, ), (2.23) .

    0

    ( )x

    t

    s t dt=

    1

    0 1

    n

    nn

    xan

    +

    = + = . (2.24) 0 ( )x

    t

    s t dt=

    2.10: 2.19(). ()

    51

    5

    1 !2 ( 5)!

    kk

    k

    k xk

    +

    = = 6 60

    1 (5 )! 5!2 ! (2 )

    rr

    r

    r xr x

    +

    =

    + = =(5)1

    2 x .

    ()

    11

    0

    1 12 1

    nn

    nx

    n

    ++

    = + = 1 2xn l = 0

    12

    x

    t

    dtt= .

  • . 2: 27

    2.1. 1:

    () =1 !

    1n n

    () () =1 !

    2n

    n

    n () ()

    = +1 )1(1

    n nn ()

    () =

    1 3

    21n

    n

    n () ()

    = +1 2 1n nn ()

    = ++

    13

    2

    234

    n nnnn

    () =1 !2n n

    n

    nn ()

    =

    +1

    2

    1n

    n

    nn () ()

    = +1 31)(

    n nn ()

    () =

    +1 2

    1)1(n

    n

    n

    () () =2 )(

    1n n nnl

    () () = +

    +1

    2 112

    n nn ()

    () =

    +1

    21

    !)1(

    n

    n

    nn () ()

    =+

    1

    1 1)1(n

    n

    n () ()

    =1 !n

    n

    nn

    () =

    1

    12

    13n

    n

    nn ()

    =1 21

    nnn

    () () =1 10

    !n

    n

    n ()

    () = 1 3 12

    )(n n

    nnl () ()

    =1

    1n

    nn ()

    =

    +1

    1)1(n

    n

    n ()

    () 2

    1 !nnn

    =

    ()

    =

    +1 13

    3n

    n

    nn ()

    ()

    = +1 3 11

    n n

    2.2.

    1

    ( )n nn

    n x

    =

    1

    1 nn

    xn

    =

    1

    1 nn

    nx

    n

    =

    () x=0

    () ..2= [1, 1)

    () .. = R

    () 1

    1 nn

    xn

    = ()

    1

    2n nn

    xn

    = () 3

    1

    12

    nn

    nx

    n

    =

    1 () : () = , () = , () = . () , , . . 2 ..= .

  • . . : -- 28

    2

    1

    12

    n

    nx

    n

    = 2

    1

    1 ( 1)1

    n

    nx

    n

    =++ 31

    1 ( 1)nn

    xn

    =

    () ..=(1, 1)

    () ..= [2, 0]

    () ..= [0, 2]

    11

    1 ( 2)2

    nn

    nx

    n

    =+

    1

    12

    n

    n

    x

    =

    2

    1( 3)

    2n

    nn

    n x

    =

    ()

    () ..= (0, 4)

    ()

    1

    1 2 ( 1)3

    nn

    nx

    n

    =

    2

    1( 1) ( 2)

    !n n

    n

    n xn

    = +

    1

    1 ( 2)nn

    xn n

    =+

    () ..= 3 3,

    2 2

    ()

    () ..= [1, 3)

  • 3

    3.1 S n (nN) . 3.1: (permutation) n . Pn

    ! 1 2 3nP n n= = K = n(n1) (n2) ... 321. (3.1)

    n! n (n factorial).

    : () (n+1)!=(n+1)n! () n!=n(n1)(n2) ...[n(r1)](nr)!= =n(n1)(n2) ...(n-r+1)(n1)! () 0!=1. 3.1: 5!=12345=120

  • . . : -- 30

    3.2

    S n (nN) rN, 0rn. 3.2: (combination) n S r .

    C(n,r),

    rn

    !( , )!( )!

    n nC n rr r n r = = . (3.2)

    C(n,r) n r.

    rn

    :

    () .

    =

    n

    rnnr

    () .

    +

    =

    11

    1 nr

    nr

    nr

    () ( =1. )0n (3.2)

    ( 1)( 2) ... ( 1)!

    n n n n n rr r += (3.3)

    nR, rN . n,rN

    1( 1)r

    n n rr r + = . (3.4)

    3.2:

    5! 5! 5 4 103!(5 3)! 3! 2! 2

    53

    5(5 1)(5 2) 5 4 3 103! 1 2 3

    = = = = = =

    3.3: :

  • . 3: 31

    14 ,

    213

    ,

    133

    ,

    134

    , 3

    2 , .

    35

    : () 1!4

    4321!4

    )31)(21)(11)(1(41 ===

    .

    () 13N.

    () 4

    1 1 1 1 4 71 ( )( 1)( 2) 14 143 3 3 3 3 333! 1 2 3 3 813

    = = = = .

    ()

    2 3 8 132 2 2 22 ( 1)( 2)( 3) 5 5 5 55 5 5 554! 1 2 3 44

    = = =

    42 3 8 13 26

    1 2 3 4 5 625 = = .

    () 2N. () 5>3. 3.3 Maclaurin

    ( ) ( )nf x a x= , ,R. :

    1. nN{}: ( ) ( )nf x a x= = 1

    nna x

    a + = 0 ( 1)

    rnn r r

    r

    n a xr a= = 0 ( 1)

    rnn r r

    r

    n a xr a= .(3.5)

    Newton.

    ( 1)r nn ar a (3.6)

    1 xr. 3.4: () 3( ) ( )f x a x= +

    1 (3.6) (1)r 1 (+) () .

  • . . : -- 32

    3( ) ( )f x a x= + =3

    3

    0

    3( 1)

    rr r

    r

    a xr a= + =

    33

    0

    3 r rr

    a xr a= =

    =0

    3 030

    x

    +1

    3 131

    x

    +2

    3 232

    x

    +3

    3 333

    x

    =

    =3 + 3 2a x +32x2 + 3x3.

    () ( ) (1 )nf x = xn n

    2( ) (1 ) ... ( 1) ... ( 1)0 1 2

    n r rn n n n nf x x x x xr n

    = = + + + + x . 2. nN: ( ) ( )nf x a x=

    ( ) ( )nf x a x= = 1n

    n a xa

    = 0 ( 1)r

    n r

    r

    n ar a

    =

    rx . (3.7) 1 (binomial series) 2

    = a

    . (3.8)

    (3.7)

    ( 1)r

    r n n ar a (3.9)

    , , xr.

    nN{}

    f(x) = ( x)n = 0

    ( 1)r

    n r

    r

    n rxr

    =

    =

    f(x) = ( x)n =0

    1( 1) ( 1)

    rn r r

    r

    n r rxr a

    =

    + . (3.10) 3.5: () f(x) = (1+x)n, nN

    f(x) = (1+x)n = 0

    11 ( 11

    rn r)

    r

    n rxr

    =

    + 0r

    r

    n = xr

    =

    =

    1 f(x)= x . . 2 .

  • . 3: 33

    = . 2 ... ...0 1 2

    rn n n nx x xr

    + + + + +

    () f(x) = (1x)n, nN,

    f(x) = (1x)n = 0

    11 ( 11

    rn r)

    r

    n rxr

    =

    0 ( 1)r rrn

    = xr

    =

    =

    = . 2 3 ... ( 1) ...0 1 2 3

    r rn n n n nx x x xr

    + + + +

    () f(x)= 1(1 )nx = (1x)

    n|x1, n,

    f(x)= ( ) ( ) ( )0 01 1 ( 1) 1

    1n rr r r

    nr r

    n nx x x

    r rx

    = =

    = = = =

    =0

    1 11 ( 1) ( 1)1

    rn r r r

    r

    n rx

    r

    =

    + 01

    ( 1) ( 1)r r rr

    n r = xr

    =

    + .

    () f(x)= 11 x = (1x)

    1|x1, 1,

    f(x)=(1x)1 =0

    1 11 ( 11

    rn r

    r) rx

    r

    =

    01 1

    ( 1) ( 1)rr

    r = r rxr

    =

    + 0r

    r

    r = xr

    =

    f(x)= 11 x = (1x)

    1=0

    r

    rx

    = =1 + x + x2 + x3 + + xr +.

    () f(x)= 11 x+ =(1+x)

    1|x1, 1,

    f(x)=(1x)1 =0

    1 11 ( 11

    rn r

    r) rx

    r

    =

    + 0 ( 1)r rrr

    = xr

    =

    f(x)= 11 x+ = (1+x)

    1=0( 1)r r

    rx

    = =1 x + x2 x3 + +(1)r xr +.

    3.6: x5 :

    () ( ) 7( ) 1 2f x x = , () 23( ) (1 )f x x= + .

    : () ( (3.10))

    7( ) (1 2 )f x x = =0

    7 2 ( 1)1

    r r

    rx

    r

    =

    = 07 1 2( 1) ( 1)

    1r r

    r

    r rxr

    =

    + .

  • . . : -- 34

    5= =14784. 5 57 5 1

    ( 1) ( 1) 25

    + 5

    ()

    23( ) (1 )f x x= + = 23(1 )x+ =( (3.7))=0

    23 r

    rx

    r

    =

    2

    35

    = 14

    729.

    3.1. :

    23

    5

    ,

    23

    5

    ,

    52

    , , , , 54

    51

    2317

    23

    45

    ,

    712 .

    ( . ). 3.2. .

    () 6( ) ( 2 5 )f x x = 9, x . () 9( ) ( 3 5 )f x x= + , 6x .

    () ( ) 1 | 1f x x x= + > , 8x , () 35

    1( )(1 2 )

    f xx

    = |12

    x > , 7x .

    (.: () 1955078125163844

    , () 35437500, () 42932768

    , () 15155712390625

    )

    3.3. (3.5) (3.6)

    ( ) ( )nf x ax y= .

    : 6 2x y

    8( ) (2 5 )f x x y= . (.: f(x)= . x

    0( 1)

    nr n r r n r

    r

    na x y

    r

    =

    rr

    nryr

    ( 1)r n rn

    ar

    . : 44800).

  • 35

    1. . .: 1999, , . , . 2. .: 1980, - - Laplace, . 3. . .: 1975, , . 4. , : 1996, , , - 5. . .- . .: 1984, , . 6. , .- .- .: 1994, , . 7. , ., . 2006, & , . - , /. 1. Agnew, R.: Calculus Mc Graw Hill " 2. APOSTOL T.: 1965, Calculus, Blaisdel Publ. Co. . 3. AYRES, F.: 1983, ( . . ) Schaum's outline series , . 4. Ayres, F.: Matrices Mc Graw Hill, 1962 5. BERMAN. G. N.: 1965, A collection of Problems on a course of Mathematical Analysis, Pergamon Press. 6. BOWMAN F.-GERARD G.: 1967, Higher Calculus, Cambridge Univ. Press. 7. BRAND Louis: 1962, Advanced Calculus, John Wiley 8. BUNDAY. B. B.-MULHOLLAND.H.: 1972, Pure Mathematics for advanced level, Butter Worths, London. 9. CAY H. J.: 1950, Analytic Geometry and Calculus, Mc Graw-Hill. 10. CHIRWIN B.-PLUMPTON C. : 1972, A course of Mathematics for Ingineers and Scientists, Pergamon Press, Oxford. 11. COURANT R.-JOHN F.: 1974, Introduction to Calculus and Analysis Wiley

    Int. . 12. DEMIDOVITCH B.: 1973, Problems in Mathematical Analysis, Moscow. 13. FOBS M. P.-SMYTH R. B.: 1963, Calculus and Analytic Geometry, Prentice Hall.

  • . . : -- 36

    14. HEGARTY C. J.: 1990, Applied Calculus, John Wiley. 15. MUNROE M.E.: 1970, Calculus, W. . Sawnders. 16. Piscunov, N.: 1974 Differential and Integral Calculus Mir Publishers. 17. RANKIN R.: 1965, An Introduction to Mathematical Analysis, Pergamon Press. 18. RUDIN W.: 1976, Principles of Mathematical Analysis, Mc Grow- Hill. 19. SALAS S.-HILLE E.: 1974, Calculus, Xerox Pub. Co. Lexington. Spiegel M.: Advanced Calculus McGraw-Hill, 1963. 20. STRANG, G: 1996, Linear algebra and its applications, ( - ) . 21. THOMAS G. B- FINNEY. R. L.: 1984, Calculus and Analytic Geometry, Addison- Wesley. 22. WHITTAKER. E. T.- WATSON. G. N.: 1965, A course of Modern Analy Harvard Univ. Press.

    - 2008