常见麦克劳林公式
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e x=
∞
∑
n=0
1
n!
x n=1+x+
1
2!
x2+···+
1
n!
x n+···,x∈(−∞,+∞)
e sin x=1+x+1
2
x2−
1
8
x4−
1
15
x5−
1
240
x6+
1
90
x7+
31
5760
x8+
1
5670
x9+o
(
x9
)
e tan x=1+x+1
2
x2+
1
2
x3+
3
8
x4+
37
120
x5+
59
240
x6+
137
720
x7+
871
5760
x8+
41641
362880
x9+o
(
x9
)
sin x=
∞
∑
n=0
(−1)n
(2n+1)!
x2n+1=x−
1
3!
x3+
1
5!
x5−···+
(−1)n
(2n+1)!
x2n+1+···,x∈(−∞,+∞)
cos x=
∞
∑
n=0
(−1)n
(2n)!
x2n=1−
1
2!
x2+
1
4!
x4−···+
(−1)n
(2n)!
x2n+···,x∈(−∞,+∞)
ln(1+x)=
∞
∑
n=0
(−1)n
n+1
x n+1=x−
1
2
x2+
1
3
x3−···+
(−1)n
n+1
x n+1+···,x∈(−1,1]
ln Å
1+x
1−x
ã
=
∞
∑
n=1
2x2n−1
2n−1
=2x+
2
3
x3+
2
5
x5+
2
7
x7+
2
9
x9+o(x9),x∈(−1,1) 1
1−x =
∞
∑
n=0
x n=1+x+x2+x3+···+x n+···,x∈(−1,1)
(1+x)12=1+1
2
x−
1
8
x2+
1
16
x3−
5
128
x4+
7
256
x5−
21
1024
x6+
33
2048
x7−
429
32768
x8+o(x8),x∈(−1,1)
(1+x)−12=1−1
2
x+
3
8
x2−
5
16
x3+
35
128
x4−
63
256
x5+
231
1024
x6−
429
2048
x7+
6435
32768
x8−
12155
65536
x9+o(x9),x∈(−1,1)
(1+x)13=1+1
3
x−
1
9
x2+
5
81
x3−
10
243
x4+
22
729
x5−
154
6561
x6+
374
19683
x7−
935
59049
x8+o
(
x8
)
,x∈(−1,1)
(1+x)−13=1−1
3
x+
2
9
x2−
14
81
x3+
35
243
x4−
91
729
x5+
728
6561
x6−
1976
19683
x7+
5453
59049
x8−
135850
1594323
x9+o
(
x9
)
,x∈(−1,1)
(1+x)32=1+3
2
x+
3
8
x2−
1
16
x3+
3
128
x4−
3
256
x5+
7
1024
x6−
9
2048
x7+
99
32768
x8−
143
65536
x9+o
(
x9
)
,x∈(−1,1)
(1+x)−32=1−3
2
x+
15
8
x2−
35
16
x3+
315
128
x4−
693
256
x5+
3003
1024
x6−
6435
2048
x7+
109395
32768
x8−
230945
65536
x9+o(x9),x∈(−1,1)
(1+x)−2=1−2x+3x2−4x3+5x4−6x5+7x6−8x7+9x8−10x9+o(x9),x∈(−1,1)
tan x=
∞
∑
n=1
B2n(−4)n(1−4n)
(2n)!
x2n−1=x+
1
3
x3+
2
15
x5+
17
315
x7+···+
(−1)n−122n(22n−1)B2n
(2n)!
x2n−1+···
Ç
x2<
π2
4
å
sec x=
∞
∑
n=0
(−1)n E2n x2n
(2n)!
=1+
1
2
x2+
5
24
x4+
61
720
x6+
277
8064
x8+o(x8)
arctan x=
∞
∑
n=0
(−1)n
2n+1
x2n+1=x−
1
3
x3+
1
5
x5+···+
(−1)n
2n+1
x2n+1+···,x∈[−1,1]
arcsin x=
∞
∑
n=0
(2n)!
4n(n!)2(2n+1)
x2n+1=x+
1
6
x3+
3
40
x5+
5
112
x7+
35
1152
x9+o(x9),x∈(−1,1)
sinh x=
∞
∑
n=0
x2n+1
(2n+1)!
=x+
x3
3!
+
x5
5!
+
x7
7!
+···+
x2n+1
(2n+1)!
+···
cosh x=
∞
∑
n=0
x2n
(2n)!
=1+
x2
2!
+
x4
4!
+
x6
6!
+···+
x2n
(2n)!
+···
tanh x=
∞
∑
n=1
22n(22n−1)B2n x2n−1
(2n)!
=x−
1
3
x3+
2
15
x5−
17
315
x7+
62
2835
x9+o(x9),|x|<
π
2
sech x=
∞
∑
n=0
E2n x2n
(2n)!
=1−
1
2
x2+
5
24
x4−
61
720
x6+
1385
40320
x8−···+
E2n
(2n)!
x2n+···,(|x|<
π
2
)
arsinh x=
∞
∑
n=0
Ç
(−1)n(2n)!
22n(n!)2
å
x2n+1
(2n+1)
=x−
1
6
x3+
3
40
x5−
5
112
x7+
35
1152
x9+o(x9),|x|<1
artanh x=
∞
∑
n=0
x2n+1
2n+1
=x+
x3
3
+
x5
5
+
x7
7
+···+
x2n+1
2n+1
+···,(|x|<1)
常见麦克劳林公式