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§5. òÑÄÙ ôÅÊÌÏÒÁ. òÁÚÌÏÖÅÎÉÅ ÆÕÎËÃÉÉ × ÓÔÅÐÅÎÎÏÊ ÒÑÄ 37
äÌÑ ÒÁÚÌÏÖÅÎÉÑ ËÏÎËÒÅÔÎÏÊ ÆÕÎËÃÉÉ f (x) × ÓÔÅÐÅÎÎÏÊ ÒÑÄ Ó ÃÅÎÔÒÏÍ ×
ÔÏÞËÅ x
0
= 0 ÐÏÌØÚÕÀÔÓÑ ÒÁÚÌÏÖÅÎÉÑÍÉ ÏÓÎÏ×ÎÙÈ ÆÕÎËÃÉÊ. ðÏÓÌÅ ËÁÖÄÏÊ
ÆÏÒÍÕÌÙ ÕËÁÚÁÎÏ ÍÎÏÖÅÓÔ×Ï ÓÈÏÄÉÍÏÓÔÉ ÒÑÄÁ.
e
x
=
∞
P
n=0
x
n
n!
= 1 + x +
x
2
2!
+
x
3
3!
+ . . . +
x
n
n!
+ . . . , |x| < ∞,
sin x =
∞
P
n=0
(−1)
n
x
2n+1
(2n+1)!
= x −
x
3
3!
+
x
5
5!
− . . . + (−1)
n
x
2n+1
(2n+1)!
+ . . . ,
|x| < ∞,
cos x =
∞
P
n=0
(−1)
n
x
2n
(2n)!
= 1 −
x
2
2!
+
x
4
4!
− . . . + (−1)
n
x
2n
(2n)!
+ . . . , |x| < ∞,
sh x =
∞
P
n=0
x
2n+1
(2n+1)!
= x +
x
3
3!
+
x
5
5!
+ . . . +
x
2n+1
(2n+1)!
+ . . . , |x| < ∞,
ch x =
∞
P
n=0
x
2n
(2n)!
= 1 +
x
2
2!
+
x
4
4!
+ . . . +
x
2n
(2n)!
+ . . . , |x| < ∞,
ln(1 + x) =
∞
P
n=1
(−1)
n−1
x
n
n
= x −
x
2
2
+
x
3
3
− . . . + (−1)
n−1
x
n
n
+ . . . ,
−1 < x 6 1,
ln(1 − x) = −
∞
P
n=1
x
n
n
= −
x +
x
2
2
+
x
3
3
+ . . . +
x
n
n
+ . . .
, −1 6 x < 1,
(1 + x)
m
= 1 +
∞
P
n=1
m(m−1)(m−2)...(m−n+1)
n!
x
n
= 1 + mx +
m(m−1)
2!
x
2
+
+
m(m−1)(m−2)
3!
x
3
+ . . . +
m(m−1)(m−2)...(m−n+1)
n!
x
n
+ . . . ,
m ∈ R, |x| < 1.
ðÒÉ×ÅÄÅÍ ÎÅËÏÔÏÒÙÅ ÞÁÓÔÎÙÅ ÓÌÕÞÁÉ ÐÏÓÌÅÄÎÅÊ ÆÏÒÍÕÌÙ.
1
1+x
= 1 − x + x
2
− x
3
+ . . . + (−1)
n
x
n
+ . . . , |x| < 1,
1
1−x
= 1 + x + x
2
+ x
3
+ . . . + x
n
+ . . . , |x| < 1,
1
√
1+x
= 1 −
1
2
x +
1·3
2·4
x
2
− . . . + (−1)
n
(2n−1)!!
(2n)!!
x
n
+ . . . , |x| < 1,
1
√
1−x
= 1 +
1
2
x +
1·3
2·4
x
2
+ . . . +
(2n−1)!!
(2n)!!
x
n
+ . . . , |x| < 1,
√
1 + x = 1 +
1
2
x −
1
2·4
x
2
+ . . . + (−1)
n
(2n−1)!!
(2n+2)!!
x
n+1
+ . . . , |x| < 1,
√
1 − x = 1 −
1
2
x −
1
2·4
x
2
− . . . −
(2n−1)!!
(2n+2)!!
x
n+1
− . . . , |x| < 1.
îÁÐÏÍÎÉÍ, ÞÔÏ ÆÁËÔÏÒÉÁÌ ÎÁÔÕÒÁÌØÎÏÇÏ ÞÉÓÌÁ n ÏÐÒÅÄÅÌÑÅÔÓÑ ÆÏÒÍÕÌÏÊ
n! = n · (n − 1) · (n − 2) · . . . ·3 · 2 · 1,
ÎÁÐÒÉÍÅÒ, 7! = 7 · 6 · 5 · 4 · 3 · 2 · 1.
ä×ÏÊÎÏÊ ÆÁËÔÏÒÉÁÌ ÞÉÓÌÁ n ÏÐÒÅÄÅÌÑÅÔÓÑ ÓÌÅÄÕÀÝÉÍ ÏÂÒÁÚÏÍ
n!! = n · (n − 2) · (n − 4) · . . . ,
ÎÁÐÒÉÍÅÒ, 7!! = 7 · 5 · 3 · 1, 10!! = 10 · 8 · 6 · 4 · 2.
§5. òÑÄÙ ôÅÊÌÏÒÁ. òÁÚÌÏÖÅÎÉÅ ÆÕÎËÃÉÉ × ÓÔÅÐÅÎÎÏÊ ÒÑÄ 37
äÌÑ ÒÁÚÌÏÖÅÎÉÑ ËÏÎËÒÅÔÎÏÊ ÆÕÎËÃÉÉ f (x) × ÓÔÅÐÅÎÎÏÊ ÒÑÄ Ó ÃÅÎÔÒÏÍ ×
ÔÏÞËÅ x0 = 0 ÐÏÌØÚÕÀÔÓÑ ÒÁÚÌÏÖÅÎÉÑÍÉ ÏÓÎÏ×ÎÙÈ ÆÕÎËÃÉÊ. ðÏÓÌÅ ËÁÖÄÏÊ
ÆÏÒÍÕÌÙ ÕËÁÚÁÎÏ ÍÎÏÖÅÓÔ×Ï ÓÈÏÄÉÍÏÓÔÉ ÒÑÄÁ.
∞ n
x x x2 x3 xn
P
e = n! = 1 + x + 2! + 3! + . . . + n! + . . . , |x| < ∞,
n=0
∞ 2n+1
x x3 x5 x 2n+1
(−1)n (2n+1)! − . . . + (−1)n (2n+1)!
P
sin x = =x− 3! + 5! + ...,
n=0
|x| < ∞,
∞ 2n 2 4 2n
x x x x
(−1)n (2n)! − . . . + (−1)n (2n)!
P
cos x = =1− 2!
+ 4!
+ ..., |x| < ∞,
n=0
∞
x2n+1 x3 x5 x2n+1
P
sh x = (2n+1)! =x+ 3! + 5! + ...+ (2n+1)! + ..., |x| < ∞,
n=0
∞
x2n x2 x4 x2n
P
ch x = (2n)! =1+ 2! + 4! + ...+ (2n)! + ..., |x| < ∞,
n=0
∞ n
x2 x3 n
(−1)n−1 xn = x − − . . . + (−1)n−1 xn + . . . ,
P
ln(1 + x) = 2
+ 3
n=1
−1 < x 6 1,
∞
xn x2 x3 xn
P
ln(1 − x) = − n =− x+ 2 + 3 + ...+ n + ... , −1 6 x < 1,
n=1
∞
m(m−1)(m−2)...(m−n+1) n m(m−1) 2
(1 + x)m = 1 +
P
n! x = 1 + mx + 2! x+
n=1
+ m(m−1)(m−2)
3! x3 + . . . + m(m−1)(m−2)...(m−n+1) n
n! x + ...,
m ∈ R, |x| < 1.
ðÒÉ×ÅÄÅÍ ÎÅËÏÔÏÒÙÅ ÞÁÓÔÎÙÅ ÓÌÕÞÁÉ ÐÏÓÌÅÄÎÅÊ ÆÏÒÍÕÌÙ.
1
1+x = 1 − x + x2 − x3 + . . . + (−1)nxn + . . . , |x| < 1,
1
1−x = 1 + x + x2 + x3 + . . . + xn + . . . , |x| < 1,
√1
1+x
= 1 − 21 x + 1·3
2·4
x2 − . . . + (−1)n (2n−1)!!
(2n)!!
xn + . . . , |x| < 1,
(2n−1)!! n
√1 = 1 + 21 x + 1·3 2
+ ...+
2·4 x (2n)!! x + . . . , |x| < 1,
√ 1−x (2n−1)!!
1 + x = 1 + 21 x − 1 2
2·4 x + . . . + (−1)n (2n+2)!! xn+1 + . . . , |x| < 1,
√
1 − x = 1 − 21 x − 1 2
2·4 x − . . . − (2n−1)!!
(2n+2)!! x
n+1
− . . . , |x| < 1.
îÁÐÏÍÎÉÍ, ÞÔÏ ÆÁËÔÏÒÉÁÌ ÎÁÔÕÒÁÌØÎÏÇÏ ÞÉÓÌÁ n ÏÐÒÅÄÅÌÑÅÔÓÑ ÆÏÒÍÕÌÏÊ
n! = n · (n − 1) · (n − 2) · . . . · 3 · 2 · 1,
ÎÁÐÒÉÍÅÒ, 7! = 7 · 6 · 5 · 4 · 3 · 2 · 1.
ä×ÏÊÎÏÊ ÆÁËÔÏÒÉÁÌ ÞÉÓÌÁ n ÏÐÒÅÄÅÌÑÅÔÓÑ ÓÌÅÄÕÀÝÉÍ ÏÂÒÁÚÏÍ
n!! = n · (n − 2) · (n − 4) · . . . ,
ÎÁÐÒÉÍÅÒ, 7!! = 7 · 5 · 3 · 1, 10!! = 10 · 8 · 6 · 4 · 2.
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