1 x = ∑ n = 0 ∞ ( − 1 ) n ( x − 1 ) n on ( 0 , 2 ) {\displaystyle {\frac {1}{x}}=\sum _{n=0}^{\infty }\left(-1\right)^{n}\!(x-1)^{n}~~{\text{ on }}~\left(0,2\right)} 1 1 + x = ∑ n = 0 ∞ ( − 1 ) n x n on ( − 1 , 1 ) {\displaystyle {\frac {1}{1+x}}=\sum _{n=0}^{\infty }\left(-1\right)^{n}\!x^{n}~~{\text{ on }}~\left(-1,1\right)} ln x = ∑ n = 1 ∞ ( − 1 ) n + 1 ( x − 1 ) n n on ( 0 , 2 ] {\displaystyle \ln x=\sum _{n=1}^{\infty }{\frac {\left(-1\right)^{n+1}\!(x-1)^{n}}{n}}~~{\text{ on }}~(0,2]} e x = ∑ n = 0 ∞ x n n ! on ( − ∞ , ∞ ) {\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}~~{\text{ on }}~\left(-\infty ,\infty \right)} sin x = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! on ( − ∞ , ∞ ) {\displaystyle \sin x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}~~{\text{ on }}~\left(-\infty ,\infty \right)} cos x = ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n ) ! on ( − ∞ , ∞ ) {\displaystyle \cos x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}~~{\text{ on }}~\left(-\infty ,\infty \right)} arctan x = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 2 n + 1 on [ − 1 , 1 ] {\displaystyle \arctan x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{2n+1}}~~{\text{ on }}~\left[-1,1\right]} arcsin x = ∑ n = 0 ∞ ( 2 n ) ! x 2 n + 1 ( 2 n n ! ) 2 ( 2 n + 1 ) on [ − 1 , 1 ] {\displaystyle \arcsin x=\sum _{n=0}^{\infty }{\frac {(2n)!~x^{2n+1}}{\left(2^{n}n!\right)^{2}\left(2n+1\right)}}~~{\text{ on }}~\left[-1,1\right]} ( 1 + x ) k = ∑ n = 0 ∞ x n product thingy n ! {\displaystyle \left(1+x\right)^{k}=\sum _{n=0}^{\infty }{\frac {x^{n}{\text{ product thingy }}}{n!}}} 1 x {\displaystyle {\frac {1}{x}}} ∑ n = 0 ∞ ( − 1 ) n ( x − 1 ) n 0 < x < 2 {\displaystyle \sum _{n=0}^{\infty }\left(-1\right)^{n}\!(x-1)^{n}\quad \qquad ~~\qquad ~0<x<2} 1 1 + x {\displaystyle {\frac {1}{1+x}}} ∑ n = 0 ∞ ( − 1 ) n x n − 1 < x < 1 {\displaystyle \sum _{n=0}^{\infty }\left(-1\right)^{n}\!x^{n}\quad \qquad \qquad \qquad -1<x<1} ln x {\displaystyle \ln x~\!} ∑ n = 1 ∞ ( − 1 ) n + 1 ( x − 1 ) n n 0 < x ≤ 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {\left(-1\right)^{n+1}\!(x-1)^{n}}{n}}\quad \quad ~\qquad ~0<x\leq 2} e x {\displaystyle e^{x}~\!} ∑ n = 0 ∞ x n n ! − ∞ < x < ∞ {\displaystyle \sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}\qquad \qquad \quad \quad ~~~\qquad -\infty <x<\infty } sin x {\displaystyle \sin x~\!} ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! − ∞ < x < ∞ {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}\quad \quad \quad \qquad -\infty <x<\infty } cos x {\displaystyle \cos x~\!} ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n ) ! − ∞ < x < ∞ {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}\qquad \quad \quad \qquad -\infty <x<\infty } arctan x {\displaystyle \arctan x~\!} ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 2 n + 1 − 1 ≤ x ≤ 1 {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{2n+1}}\quad \quad \quad ~~\qquad -1\leq x\leq 1} arcsin x {\displaystyle \arcsin x~\!} ∑ n = 0 ∞ ( 2 n ) ! x 2 n + 1 ( 2 n n ! ) 2 ( 2 n + 1 ) − 1 ≤ x ≤ 1 {\displaystyle \sum _{n=0}^{\infty }{\frac {(2n)!~x^{2n+1}}{\left(2^{n}n!\right)^{2}\left(2n+1\right)}}~~\qquad \quad -1\leq x\leq 1} ( 1 + x ) k {\displaystyle \left(1+x\right)^{k}} 1 + k x + k ( k − 1 ) x 2 2 ! + k ( k − 1 ) ( k − 2 ) x 3 3 ! + ⋯ {\displaystyle 1+kx+{\frac {k(k-1)x^{2}}{2!}}+{\frac {k(k-1)(k-2)x^{3}}{3!}}+\cdots } 1 + k x + k ( k − 1 ) x 2 2 ! + k ( k − 1 ) ( k − 2 ) x 3 3 ! + ⋯ − 1 < x < 1 ; x may equal ± 1 {\displaystyle {\begin{aligned}&1+kx+{\frac {k(k-1)x^{2}}{2!}}+{\frac {k(k-1)(k-2)x^{3}}{3!}}+\cdots \\&\qquad ~~-1<x<1\quad {\text{;}}\quad x{\text{ may equal}}\pm 1\\\end{aligned}}}
![{\displaystyle \ln x=\sum _{n=1}^{\infty }{\frac {\left(-1\right)^{n+1}\!(x-1)^{n}}{n}}~~{\text{ on }}~(0,2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fed92666922f599b11c2fef67028424e3717486a)
![{\displaystyle \arctan x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{2n+1}}~~{\text{ on }}~\left[-1,1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65a413a99298762e2bd233f07193c51653c2b6a1)
![{\displaystyle \arcsin x=\sum _{n=0}^{\infty }{\frac {(2n)!~x^{2n+1}}{\left(2^{n}n!\right)^{2}\left(2n+1\right)}}~~{\text{ on }}~\left[-1,1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4075ef7fcd0b5921ee5efc2daee58b399501d4)
