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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