Maclaurin Series

A series expansion of a function about 0,

$\begin{displaymath} f(x) = f(0)+f'(0)x+{f''(0)\over 2!} x^2+{f^{(3)}(0)\over 3!} x^3+\ldots +{f^{(n)}(0)\over n!} x^n+\ldots, \end{displaymath}$

(1)

named after the Scottish mathematician Maclaurin.

Maclaurin series for common functions include

${1\over 1-x} = 1+x+x^2+x^3+x^4+x^5+\ldots{\rm\ for\ } -1 < x < 1$ (2)

$\mathop{\rm cn}\nolimits (x,k^2) =1-{\textstyle{1\over 2!}}x^2+{\textstyle{1\over 4!}}(1+4k^2)x^4+\ldots$ (3)

$\cos x = 1-{\textstyle{1\over 2!}}x^2+{\textstyle{1\over 4!}}x^4-{\textstyle{1\over 6!}}x^6-\ldots{\rm\ for\ } -\infty <x<\infty$ (4)

$\cos^{-1} x = {\textstyle{1\over 2}}\pi-x-{\textstyle{1\over 6}}x^3-{\textstyle{3\over 40}}x^5-{\textstyle{5\over 112}}x^7-\ldots {\rm\ for\ } -1 < x < 1$ (5)

$\cosh x = 1+{\textstyle{1\over 2}}x^2+{\textstyle{1\over 24}}x^4+{\textstyle{1\over 720}}x^6+{\textstyle{1\over 40{},320}}x^8+\ldots$ (6)

$\cosh^{-1} (1+x) =\sqrt{2x}\, (1-{\textstyle{1\over 2}}x+{\textstyle{3\over 160}} x^2-{\textstyle{5\over 896}} x^3+\ldots)$ (7)

$\cot x = x^{-1}-{\textstyle{1\over 3}}x-{\textstyle{1\over 45}}x^3-{\textstyle{2\over 945}}x^5-{\textstyle{1\over 4725}}x^7-\ldots$ (8)

$\cot^{-1}x = {\textstyle{1\over 2}}\pi-x+{\textstyle{1\over 3}}x^3-{\textstyle{1\over 5}}x^5+{\textstyle{1\over 7}}x^7-{\textstyle{1\over 9}}x^9+\ldots$ (9)

$\phantom{\cot^{-1}x} = x^{-1}-{\textstyle{1\over 3}}x^{-3}+{\textstyle{1\over 5}}x^{-5}-{\textstyle{1\over 7}}x^{-7}+{\textstyle{1\over 9}}x^{-9}+\ldots$ (10)

$\coth x =x^{-1}+{\textstyle{1\over 3}}x-{\textstyle{1\over 45}} x^4+{\textstyle{2\over 945}} x^5-{\textstyle{1\over 4725}}x^7+\ldots$ (11)

$\coth^{-1} (1+x) = {\textstyle{1\over 2}}\ln 2-{\textstyle{1\over 2}}\ln x+{\textstyle{1\over 4}}x-{\textstyle{1\over 16}}x^2+\ldots$ (12)

$\csc x =x^{-1}+{\textstyle{1\over 6}}x+{\textstyle{7\over 360}}x^3+{\textstyle{31\over 15120}}x^5+\ldots$ (13)

$\mathop{\rm csch}\nolimits x = x^{-1} -{\textstyle{1\over 6}} x+{\textstyle{7\over 360}}x^3+{\textstyle{31\over 15120}}x^5+\ldots$ (14)

$\mathop{\rm csch}\nolimits ^{-1} x =\ln 2-\ln x+{\textstyle{1\over 4}}x^2-{\textstyle{3\over 32}} x^4 +{\textstyle{5\over 96}} x^6-\ldots$ (15)

$\mathop{\rm dn}\nolimits (x,k^2) x =1-{\textstyle{1\over 2!}} k^2x^2+{\textstyle{1\over 4!}}k^2(4+k^2)x^4+\ldots$ (16)

$\mathop{\rm erf}\nolimits x = {1\over\sqrt{\pi}}\left({2x-{\textstyle{2\over 3}} x^3+{\textstyle{1\over 5}} x^5 -{\textstyle{1\over 21}} x^7+\ldots}\right)$ (17)

$e^x = 1+x+{\textstyle{1\over 2!}}x^2+{\textstyle{1\over 3!}}x^3+{\textstyle{1\over 4!}}x^4 +\ldots {\rm\ for\ } -\infty<x<\infty$ (18)

${}_2F_1(\alpha,\beta,\gamma ;x) = 1 + {\alpha\beta\over 1!\gamma} x + {\alpha(\alpha+1)\beta(\beta+1)\over 2!\gamma(\gamma+1)} x^2 + \ldots$ (19)

$\ln(1+x) = x-{\textstyle{1\over 2}}x^2 +{\textstyle{1\over 3}}x^3-{\textstyle{1\over 4}}x^4+\ldots{\rm\ for\ } -1 < x < 1$ (20)

$\ln\left({1+x\over 1-x}\right)= 2x+{\textstyle{2\over 3}}x^3+{\textstyle{2\over 5}}x^5+{\textstyle{2\over 7}}x^7+\ldots {\rm\ for\ } -1 < x < 1$ (21)

$\sec x =1+{\textstyle{1\over 2}}x^2+{\textstyle{5\over 24}}x^4+{\textstyle{61\over 720}}x^6+{\textstyle{277\over 8064}}x^8+\ldots$ (22)

$\mathop{\rm sech}\nolimits x = 1-{\textstyle{1\over 2}}x^2+{\textstyle{5\over 24}}x^4-{\textstyle{61\over 720}}x^6+{\textstyle{277\over 8064}}x^8+\ldots$ (23)

$\mathop{\rm sech}\nolimits ^{-1} x =\ln 2-\ln x-{\textstyle{1\over 4}}x^2-{\textstyle{3\over 32}} x^4 -\ldots$ (24)

$\sin x = x-{\textstyle{1\over 3!}}x^3+{\textstyle{1\over 5!}}x^5-{\textstyle{1\over 7!}}x^7+\ldots{\rm\ for\ } -\infty<x<\infty$ (25)

$\sin^{-1} x = x+{\textstyle{1\over 6}}x^3+{\textstyle{3\over 40}}x^5+{\textstyle{5\over 112}}x^7+{\textstyle{35\over 1152}}x^9+\ldots$ (26)

$\sinh x = x+{\textstyle{1\over 6}}x^3+{\textstyle{1\over 120}} x^5+{\textstyle{1\over 5040}}x^7+{\textstyle{1\over 362{,}880}} x^9+ \ldots$ (27)

$\sinh^{-1} x = x-{\textstyle{1\over 6}}x^3+{\textstyle{3\over 40}}x^5-{\textstyle{5\over 112}}x^7+{\textstyle{35\over 1152}}x^9-\ldots$ (28)

$\mathop{\rm sn}\nolimits (x,k^2)={\textstyle{1\over 3!}}(1+k^2)x^3+{\textstyle{1\over 5!}}(1+14k^2+k^4)x^5+\ldots$ (29)

$\tan x = x+{\textstyle{1\over 3}}x^3+{\textstyle{2\over 15}}x^5+{\textstyle{17\over 315}}x^7+{\textstyle{62\over 2835}}x^9+\ldots$ (30)

$\tan^{-1} x = x-{\textstyle{1\over 3}}x^3+{\textstyle{1\over 5}}x^5-{\textstyle{1\over 7}}x^7+\ldots {\rm\ for\ } -1 < x < 1$ (31)

$\tan^{-1}(1+x) = {\textstyle{1\over 4}}\pi+{\textstyle{1\over 2}}x-{\textstyle{1\over 4}}x^2+{\textstyle{1\over 12}}x^3+{\textstyle{1\over 40}}x^5+\ldots$ (32)

$\tanh x = x-{\textstyle{1\over 3}}x^3+{\textstyle{2\over 15}}x^5-{\textstyle{17\over 315}}x^7+{\textstyle{62\over 2835}}x^9+\ldots$ (33)

$\tanh^{-1} x = x+{\textstyle{1\over 3}}x^3+{\textstyle{1\over 5}}x^5+{\textstyle{1\over 7}}x^7+{\textstyle{1\over 9}}x^9+\ldots.$ (34)

The explicit forms for some of these are

${1\over 1-x} = \sum_{n=0}^\infty x^n$ (35)

$\cos x = \sum_{n=0}^\infty {(-1)^n \over(2n)!}x^{2n}$ (36)

$\csc x = \sum_{n=0}^\infty {(-1)^{n+1} 2 (2^{2n-1}-1)B_{2n}\over (2n)!} x^{2n-1}$ (37)

$e^x = \sum_{n=0}^\infty {1\over n!} x^n$ (38)

$\ln(1+x) = \sum_{n=1}^\infty {(-1)^{n+1}\over n} x^n$ (39)

$\ln\left({1+x\over 1-x}\right)= \sum_{n=1}^\infty {2\over(2n-1)}x^{2n-1}$ (40)

$\sec x = \sum_{n=0}^\infty{(-1)^n E_{2n}\over (2n)!} x^{2n}$ (41)

$\sin x = \sum_{n=1}^\infty {(-1)^{n+1}\over (2n-1)!} x^{2n-1}$ (42)

$\tan x = \sum_{n=1}^\infty {(-1)^{n+1}2^{2n}(2^{2n}-1)B_{2n}\over(2n)!} x^{2n-1}$ (43)

$\tan^{-1} x = \sum_{n=1}^\infty {(-1)^{n+1}\over 2n-1}x^{2n-1}$ (44)

$\tanh^{-1} x = \sum_{n=1}^\infty {1\over 2n-1}x^{2n-1},$ (45)

where are Bernoulli Numbers and are Euler Numbers.

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 299-300, 1987.

${1\over 1-x} = 1+x+x^2+x^3+x^4+x^5+\ldots{\rm\ for\ } -1 < x < 1$	(2)
$\mathop{\rm cn}\nolimits (x,k^2) =1-{\textstyle{1\over 2!}}x^2+{\textstyle{1\over 4!}}(1+4k^2)x^4+\ldots$	(3)
$\cos x = 1-{\textstyle{1\over 2!}}x^2+{\textstyle{1\over 4!}}x^4-{\textstyle{1\over 6!}}x^6-\ldots{\rm\ for\ } -\infty <x<\infty$	(4)
$\cos^{-1} x = {\textstyle{1\over 2}}\pi-x-{\textstyle{1\over 6}}x^3-{\textstyle{3\over 40}}x^5-{\textstyle{5\over 112}}x^7-\ldots {\rm\ for\ } -1 < x < 1$	(5)
$\cosh x = 1+{\textstyle{1\over 2}}x^2+{\textstyle{1\over 24}}x^4+{\textstyle{1\over 720}}x^6+{\textstyle{1\over 40{},320}}x^8+\ldots$	(6)
$\cosh^{-1} (1+x) =\sqrt{2x}\, (1-{\textstyle{1\over 2}}x+{\textstyle{3\over 160}} x^2-{\textstyle{5\over 896}} x^3+\ldots)$	(7)
$\cot x = x^{-1}-{\textstyle{1\over 3}}x-{\textstyle{1\over 45}}x^3-{\textstyle{2\over 945}}x^5-{\textstyle{1\over 4725}}x^7-\ldots$	(8)
$\cot^{-1}x = {\textstyle{1\over 2}}\pi-x+{\textstyle{1\over 3}}x^3-{\textstyle{1\over 5}}x^5+{\textstyle{1\over 7}}x^7-{\textstyle{1\over 9}}x^9+\ldots$	(9)
$\phantom{\cot^{-1}x} = x^{-1}-{\textstyle{1\over 3}}x^{-3}+{\textstyle{1\over 5}}x^{-5}-{\textstyle{1\over 7}}x^{-7}+{\textstyle{1\over 9}}x^{-9}+\ldots$	(10)
$\coth x =x^{-1}+{\textstyle{1\over 3}}x-{\textstyle{1\over 45}} x^4+{\textstyle{2\over 945}} x^5-{\textstyle{1\over 4725}}x^7+\ldots$	(11)
$\coth^{-1} (1+x) = {\textstyle{1\over 2}}\ln 2-{\textstyle{1\over 2}}\ln x+{\textstyle{1\over 4}}x-{\textstyle{1\over 16}}x^2+\ldots$	(12)
$\csc x =x^{-1}+{\textstyle{1\over 6}}x+{\textstyle{7\over 360}}x^3+{\textstyle{31\over 15120}}x^5+\ldots$	(13)
$\mathop{\rm csch}\nolimits x = x^{-1} -{\textstyle{1\over 6}} x+{\textstyle{7\over 360}}x^3+{\textstyle{31\over 15120}}x^5+\ldots$	(14)
$\mathop{\rm csch}\nolimits ^{-1} x =\ln 2-\ln x+{\textstyle{1\over 4}}x^2-{\textstyle{3\over 32}} x^4 +{\textstyle{5\over 96}} x^6-\ldots$	(15)
$\mathop{\rm dn}\nolimits (x,k^2) x =1-{\textstyle{1\over 2!}} k^2x^2+{\textstyle{1\over 4!}}k^2(4+k^2)x^4+\ldots$	(16)
$\mathop{\rm erf}\nolimits x = {1\over\sqrt{\pi}}\left({2x-{\textstyle{2\over 3}} x^3+{\textstyle{1\over 5}} x^5 -{\textstyle{1\over 21}} x^7+\ldots}\right)$	(17)
$e^x = 1+x+{\textstyle{1\over 2!}}x^2+{\textstyle{1\over 3!}}x^3+{\textstyle{1\over 4!}}x^4 +\ldots {\rm\ for\ } -\infty<x<\infty$	(18)

${}_2F_1(\alpha,\beta,\gamma ;x) = 1 + {\alpha\beta\over 1!\gamma} x + {\alpha(\alpha+1)\beta(\beta+1)\over 2!\gamma(\gamma+1)} x^2 + \ldots$	(19)
$\ln(1+x) = x-{\textstyle{1\over 2}}x^2 +{\textstyle{1\over 3}}x^3-{\textstyle{1\over 4}}x^4+\ldots{\rm\ for\ } -1 < x < 1$	(20)
$\ln\left({1+x\over 1-x}\right)= 2x+{\textstyle{2\over 3}}x^3+{\textstyle{2\over 5}}x^5+{\textstyle{2\over 7}}x^7+\ldots {\rm\ for\ } -1 < x < 1$	(21)
$\sec x =1+{\textstyle{1\over 2}}x^2+{\textstyle{5\over 24}}x^4+{\textstyle{61\over 720}}x^6+{\textstyle{277\over 8064}}x^8+\ldots$	(22)
$\mathop{\rm sech}\nolimits x = 1-{\textstyle{1\over 2}}x^2+{\textstyle{5\over 24}}x^4-{\textstyle{61\over 720}}x^6+{\textstyle{277\over 8064}}x^8+\ldots$	(23)
$\mathop{\rm sech}\nolimits ^{-1} x =\ln 2-\ln x-{\textstyle{1\over 4}}x^2-{\textstyle{3\over 32}} x^4 -\ldots$	(24)
$\sin x = x-{\textstyle{1\over 3!}}x^3+{\textstyle{1\over 5!}}x^5-{\textstyle{1\over 7!}}x^7+\ldots{\rm\ for\ } -\infty<x<\infty$	(25)
$\sin^{-1} x = x+{\textstyle{1\over 6}}x^3+{\textstyle{3\over 40}}x^5+{\textstyle{5\over 112}}x^7+{\textstyle{35\over 1152}}x^9+\ldots$	(26)
$\sinh x = x+{\textstyle{1\over 6}}x^3+{\textstyle{1\over 120}} x^5+{\textstyle{1\over 5040}}x^7+{\textstyle{1\over 362{,}880}} x^9+ \ldots$	(27)
$\sinh^{-1} x = x-{\textstyle{1\over 6}}x^3+{\textstyle{3\over 40}}x^5-{\textstyle{5\over 112}}x^7+{\textstyle{35\over 1152}}x^9-\ldots$	(28)
$\mathop{\rm sn}\nolimits (x,k^2)={\textstyle{1\over 3!}}(1+k^2)x^3+{\textstyle{1\over 5!}}(1+14k^2+k^4)x^5+\ldots$	(29)
$\tan x = x+{\textstyle{1\over 3}}x^3+{\textstyle{2\over 15}}x^5+{\textstyle{17\over 315}}x^7+{\textstyle{62\over 2835}}x^9+\ldots$	(30)
$\tan^{-1} x = x-{\textstyle{1\over 3}}x^3+{\textstyle{1\over 5}}x^5-{\textstyle{1\over 7}}x^7+\ldots {\rm\ for\ } -1 < x < 1$	(31)
$\tan^{-1}(1+x) = {\textstyle{1\over 4}}\pi+{\textstyle{1\over 2}}x-{\textstyle{1\over 4}}x^2+{\textstyle{1\over 12}}x^3+{\textstyle{1\over 40}}x^5+\ldots$	(32)
$\tanh x = x-{\textstyle{1\over 3}}x^3+{\textstyle{2\over 15}}x^5-{\textstyle{17\over 315}}x^7+{\textstyle{62\over 2835}}x^9+\ldots$	(33)
$\tanh^{-1} x = x+{\textstyle{1\over 3}}x^3+{\textstyle{1\over 5}}x^5+{\textstyle{1\over 7}}x^7+{\textstyle{1\over 9}}x^9+\ldots.$	(34)

${1\over 1-x} = \sum_{n=0}^\infty x^n$	(35)
$\cos x = \sum_{n=0}^\infty {(-1)^n \over(2n)!}x^{2n}$	(36)
$\csc x = \sum_{n=0}^\infty {(-1)^{n+1} 2 (2^{2n-1}-1)B_{2n}\over (2n)!} x^{2n-1}$	(37)
$e^x = \sum_{n=0}^\infty {1\over n!} x^n$	(38)
$\ln(1+x) = \sum_{n=1}^\infty {(-1)^{n+1}\over n} x^n$	(39)
$\ln\left({1+x\over 1-x}\right)= \sum_{n=1}^\infty {2\over(2n-1)}x^{2n-1}$	(40)
$\sec x = \sum_{n=0}^\infty{(-1)^n E_{2n}\over (2n)!} x^{2n}$	(41)
$\sin x = \sum_{n=1}^\infty {(-1)^{n+1}\over (2n-1)!} x^{2n-1}$	(42)
$\tan x = \sum_{n=1}^\infty {(-1)^{n+1}2^{2n}(2^{2n}-1)B_{2n}\over(2n)!} x^{2n-1}$	(43)
$\tan^{-1} x = \sum_{n=1}^\infty {(-1)^{n+1}\over 2n-1}x^{2n-1}$	(44)
$\tanh^{-1} x = \sum_{n=1}^\infty {1\over 2n-1}x^{2n-1},$	(45)