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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