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ΑΚΟΛΟΥΘΙΕΣ - ΣΕΙΡΕΣΔΙΩΝΥΜΙΚΕΣ ΔΥΝΑΜΟΣΕΙΡΕΣΠεριληπτική Θεωρία και ΑσκήσειςΘεσσαλονίκη
Citation preview
.
-
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