常见麦克劳林公式大全_wrapper_wrapper

∞ n ∑ B2n (−4) (1 − 4n ) n=1 ∞ ∑
(2n)!
n
x
2n−1
1 2 17 7 (−1) = x + x3 + x5 + x + ··· + 3 15 315
n−1 2n
2
(22n − 1) B2n 2n−1 π2 x + · · · x2 < (2n)! 4
Ç
å
(−1) E2n x2n 1 5 61 6 277 8 = 1 + x2 + x4 + x + x + o(x8 ) (2 n )! 2 24 720 8064 n=0 1 1 (−1) 2n+1 x2n+1 = x − x3 + x5 + · · · + x + · · · , x ∈ [−1, 1] 2 n + 1 3 5 2 n+1 n=0
x3 x5 x7 x2n+1 x2n+1 =x+ + + + ··· + + ··· (2n + 1)! 3! 5! 7! (2n + 1)! n=0
∞ ∑ x2n n=0 ∞ ∑
∞ ∑
(2n)!
=1+
x2 x4 x6 x2n + + + ··· + + ··· 2! 4! 6! (2n)!
1 2 17 7 62 9 π 22n (22n − 1) B2n x2n−1 = x − x3 + x5 − x + x + o(x9 ), |x| < (2 n )! 3 15 315 2835 2 n=1 (2n)! 1 5 61 6 1385 8 E2n 2n π = 1 − x2 + x4 − x + x − ··· + x + · · · , (|x| < ) 2 24 720 40320 (2n)! 2
n
∞ ∑ E2n x2n n=0 ∞ Ç ∑ n=0 ∞ ∑
arsinh x =
(−1) (2n)! 22n (n!)
2
å
1 3 5 7 35 9 x2n+1 = x − x3 + x5 − x + x + o(x9 ), |x| < 1 (2n + 1) 6 40 112 1152
artanh x =
n
ñ
ô
1 1 1 1 1 + 3 + 5 + 7 + ··· + + · · · , x ∈ (|x| > 1) x 3x 5x 7x (2n + 1)x2n−1 1 1 8 7 13 9 47 11 sin(sin x) = x − x3 + x5 − x + x − x + o(x11 ) 3 10 315 2520 49896 1 1 55 7 143 9 968167 11 x − x − x + o(x11 ) sin(tan x) = x + x3 − x5 − 6 40 1008 3456 39916800 1 1 1 9 1 11 sin(sinh x) = x − x5 − x7 + x + x + o(x11 ) 15 90 5670 3150 3 5 35 9 63 11 1 sin(arctan x) = x − x3 + x5 − x7 + x − x + o(x11 ) 2 8 16 128 256 2 3 181 7 59 9 3455 11 tan(tan x) = x + x3 + x5 + x + x + x + o(x11 ) 3 5 315 105 6237 1 1 107 7 73 9 41897 11 tan(sin x) = x + x3 − x5 − x − x + x + o(x11 ) 6 40 5040 24192 39916800 1 3 5 35 9 63 11 tan(arcsin x) = x + x3 + x5 + x7 + x + x + o(x11 ) 2 8 16 128 256 arcoth x =
∞
1 1 1 5 4 7 5 21 6 33 7 429 8 1 (1 + x) 2 = 1 + x − x2 + x3 − x + x − x + x − x + o(x8 ), x ∈ (−1, 1) 2 8 16 128 256 1024 2048 32768 3 5 35 4 63 5 231 6 429 7 6435 8 12155 9 1 1 (1 + x)− 2 = 1 − x + x2 − x3 + x − x + x − x + x − x + o(x9 ), x ∈ (−1, 1) 2 8 16 128 256 1024 2048 32768 65536 1 ( ) 1 1 5 10 4 22 5 154 6 374 7 935 8 (1 + x) 3 = 1 + x − x2 + x3 − x + x − x + x − x + o x8 , x ∈ (−1, 1) 3 9 81 243 729 6561 19683 59049 1 ( ) 1 2 2 14 3 35 4 91 5 728 6 1976 7 5453 8 135850 9 −3 (1 + x) = 1 − x + x − x + x − x + x − x + x − x + o x9 , x ∈ (−1, 1) 3 9 81 243 729 6561 19683 59049 1594323 3 ( ) 3 3 1 3 3 7 9 99 143 2 3 4 5 6 7 8 (1 + x) 2 = 1 + x + x − x + x − x + x − x + x − x9 + o x9 , x ∈ (−1, 1) 2 8 16 128 256 1024 2048 32768 65536 3 15 35 315 693 3003 6435 109395 230945 9 3 x4 − x5 + x6 − x7 + x8 − x + o(x9 ), x ∈ (−1, 1) (1 + x)− 2 = 1 − x + x2 − x3 + 2 8 16 128 256 1024 2048 32768 65536 (1 + x)−2 = 1 − 2x + 3x2 − 4x3 + 5x4 − 6x5 + 7x6 − 8x7 + 9x8 − 10x9 + o(x9 ), x ∈ (−1, 1) tan x = sec x =
ln
1+x 1−x
ã
=
2x2n−1 2 2 2 2 = 2x + x3 + x5 + x7 + x9 + o(x9 ), x ∈ (−1, 1) 2 n − 1 3 5 7 9 n=1
∑ 1 = xn = 1 + x + x2 + x3 + · · · + xn + · · · , x ∈ (−1, 1) 1 − x n=0
1 1 5 1 17 6 13 7 629 8 325 9 8177 10 earcsin x = 1 + x + x2 + x3 + x4 + x5 + x + x + x + x + x + o(x10 ) 2 3 24 6 144 126 8064 4326 145152 1 1 7 1 29 6 1 7 1219 8 1163 9 17321 10 earctan x = 1 + x + x2 − x3 − x4 + x5 + x − x − x − x + x + o(x10 ) 2 6 24 24 144 1008 8064 72576 145152 x 5e 5e 13e 5 203e 6 877e 7 23e 8 1007e 9 4639e 10 ee = e + ex + ex2 + x3 + x4 + x + x + x + x + x + x + o(x10 ) 6 8 30 720 5040 244 17280 145152 2n−1 1 1 4 1 6 1 B2n 2n n2 x − x − x8 + · · · + (−1) x + · · · , 0 < x2 < π 2 ln sin x = ln x − x2 − 6 180 2835 37800 n (2n)! 2n−1 1 1 1 17 8 31 10 (22n−1 − 1) B2n 2n π2 n2 ln cos x = − x2 − x4 − x6 − x − x + · · · + (−1) x + · · · , x2 < 2 12 45 2520 14175 n (2n)! 4 2n 2n−1 2 1 7 62 6 127 8 (2 − 1) B2n 2n π n−1 2 lntanx = ln x + x2 + x4 + x + x + · · · + (−1) x + · · · , 0 < x2 < 3 90 2835 18900 n (2n)! 4 csc x = cot x =
x3 x5 x7 x2n+1 x2n+1 =x+ + + + ··· + + · · · , (|x| < 1) 2n + 1 3 5 7 2n + 1 n=0
(1 + x) = 1 +
α
∞ ∑ α (α − 1) · · · (α − n + 1) n=1
n!
xn = 1 + αx +
α (α − 1) 2 α (α − 1) ... (α − n + 1) n x + ··· + x + · · · , x ∈ (−1, 1) 2! n!
∞ ∑ 22n B2n n=0 ∞ ∑
coth x = csch x =
(2n)!
x2n−1 =
1 1 1 2 5 22n B2n 2n−1 + x − x3 + x − ··· + x − · · · , (0 < |x| < π ) x 3 45 945 (2n)!
2 (22n−1 − 1) B2n 2n−1 1 1 7 3 31 5 127 7 x = − x+ x − x + x + o(x7 ), x ∈ (0, π ) (2 n )! x 6 360 15120 604800 n=0
∞ ∑ n=1 ∞
Ç
arcosh x = ln 2x − arccos x =
(−1) (2n)! x−2n = ln 2x − 2 2n 22n (n!)
n
å
Ç
1 −2 3 15 −6 x + x−4 + x + ··· + 4 32 288
Ç
(−1) (2n)! 22n (n!)
ò
2
n
å
x−2n + · · · , |x| > 1 2n
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ex =
∞ ∑ 1 n=0
n!
xn = 1 + x +
1 2 1 x + · · · + xn + · · · , x ∈ (−∞, +∞) 2! n!
( ) 1 1 1 1 6 1 31 8 1 9 esin x = 1 + x + x2 − x4 − x5 − x + x7 + x + x + o x9 2 8 15 240 90 5760 5670 ( ) 1 1 3 37 5 59 6 137 7 871 8 41641 9 etan x = 1 + x + x2 + x3 + x4 + x + x + x + x + x + o x9 2 2 8 120 240 720 5760 362880 ∞ ∑ (−1)n 2n+1 1 3 1 5 (−1)n 2n+1 sin x = x = x − x + x − ··· + x + · · · , x ∈ (−∞, +∞) (2n + 1)! 3! 5! (2n + 1)! n=0
∞ ∑ n=0 ∞ n ∑ (−1) n
arctan x = arcsin x = sinh x = cosh x = tanh x = sech x =
(2n)!
2 4n (n!)
1 3 5 7 35 9 x2n+1 =x + x3 + x5 + x + x + o(x9 ), x ∈ (−1, 1) 6 40 112 1152 (2n + 1)
cos x =
∞ n ∑ (−1) n=0
(2n)!
x2n = 1 −
1 2 1 (−1) 2n x + x4 − · · · + x + · · · , x ∈ (−∞, +∞) 2! 4! (2n)!
n
n
ln(1 + x) =
Å
∞ n ∑ (−1) n=0 ∞ ∑
1 1 (−1) n+1 xn+1 = x − x2 + x3 − · · · + x + · · · , x ∈ (−1, 1] n+1 2 3 n+1
å
(2n)! π 1 3 5 7 35 9 π ∑ − x2n+1 = − x + x3 + x5 + x + x + o(x9 ) , 2 n=0 4n (n!)2 (2n + 1) 2 6 40 112 1152
∞ n
ï
( ) π ∑ (−1) 2n+1 π 1 1 (−1) 2n+1 arccot x = − x = − x − x3 + x5 + · · · + x + · · · , x2 < 1 2 n=0 2n + 1 2 3 5 2n + 1
∞ n+1 ∑ (−1) 2 (22n−1 − 1) B2n n=0 ∞ ∑
(2n)!
n 2n
x2n−1 =
1 1 7 3 31 5 127 7 + x+ x + x + x + o(x7 ), x ∈ (0, π ) x 6 360 15120 604800
(−1) 2 B2n 2n−1 1 1 1 2 5 1 7 x = − x − x3 − x − x + o(x7 ), x ∈ (0, π ) (2 n )! x 3 45 945 4725 n=0