User:Humanist Geek/power series

$$ \frac{1}{x} = \sum_{n=0}^\infty \left( -1 \right)^n \!(x-1)^n \text{ on }~ \left( 0,2 \right) $$

$$ \frac{1}{1+x} = \sum_{n=0}^\infty \left( -1 \right)^n \! x^n \text{ on }~ \left( -1,1 \right) $$

$$ \ln x = \sum_{n=1}^\infty \frac{\left( -1 \right)^{n+1} \! (x-1)^n}{n} \text{ on }~ (0,2] $$

$$ e^x = \sum_{n=0}^\infty \frac{x^n}{n!} \text{ on }~ \left( -\infty, \infty \right)$$

$$ \sin x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} \text{ on }~ \left( -\infty, \infty \right) $$

$$ \cos x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} \text{ on }~ \left( -\infty, \infty \right) $$

$$ \arctan x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{2n+1} \text{ on }~ \left[ -1, 1 \right] $$

$$ \arcsin x = \sum_{n=0}^\infty \frac{(2n)!~x^{2n+1}}{ \left( 2^{n}n! \right)^{2} \left( 2n+1 \right) } \text{ on }~ \left[ -1, 1 \right] $$

$$ \left( 1+x \right)^{k} = \sum_{n=0}^\infty \frac{x^{n} \text{ product thingy }}{n!} $$

$$ \frac{1}{x} $$ $$ \sum_{n=0}^\infty \left( -1 \right)^n \!(x-1)^n \quad\qquad \qquad~ 0 < x < 2 $$

$$ \frac{1}{1+x} $$ $$ \sum_{n=0}^\infty \left( -1 \right)^n \! x^n \quad\qquad \qquad \qquad -1< x <1 $$

$$ \ln x ~\!$$ $$ \sum_{n=1}^\infty \frac{\left( -1 \right)^{n+1} \! (x-1)^n}{n} \quad\quad ~\qquad~ 0< x \leq 2 $$

$$ e^x ~\!$$ $$ \sum_{n=0}^\infty \frac{x^n}{n!} \qquad \qquad \quad\quad \qquad -\infty < x < \infty $$

$$ \sin x ~\!$$ $$ \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} \quad\quad \quad \qquad -\infty < x < \infty $$

$$ \cos x ~\!$$ $$ \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} \qquad \quad\quad \qquad -\infty < x < \infty $$

$$ \arctan x ~\!$$ $$ \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{2n+1} \quad \quad\quad  \qquad -1 \leq x \leq 1 $$

$$ \arcsin x ~\!$$ $$ \sum_{n=0}^\infty \frac{(2n)!~x^{2n+1}}{ \left( 2^{n}n! \right)^{2} \left( 2n+1 \right) } \qquad \quad-1 \leq x \leq 1 $$

$$ \left( 1+x \right)^{k} $$ $$ 1 + kx + \frac{k(k-1)x^2}{2!} +\frac{k(k-1)(k-2)x^3}{3!} + \cdots $$

$$ \begin{align} & 1 + kx + \frac{k(k-1)x^2}{2!} +\frac{k(k-1)(k-2)x^3}{3!} + \cdots \\ & \qquad -1 < x < 1 \quad \text{;} \quad x\text{ may equal} \pm 1 \\ \end{align}$$