Power series

1. Geometric series
   
   $$\sum_0^\infty ax^i = a+ ax + ax^2 +\cdots = \frac{a}{1-x} \mbox{ where $\vert x\vert<1$ }.$$
   
   

2. Exponential series
   
   $$\exp(x) = \sum_0^\infty x^i = 1+ x + \frac{x^2}{2!}+\cdots +\frac{x^n}{n!}+\cdots \mbox{ for all $x\in \Re$ }.$$
   
   

3. Basic Trigonometric series

   
   

   $$\sin(x) = \sum_0^\infty (-1)^i\frac{x^{2i+1}}{(2i+1)!} = x -... ...x^3}{3!} + \frac{x^5}{5!} - \cdots \mbox{ for all $x\in \Re$ }.$$
   
   

4. Generalized Binomial series
   
   $$(1+x)^n = \sum_0^\infty {}_nC_i x^i = \sum_0^\infty \frac{(n)(n-1)\cdots (n-i+1)}{i!} x^i$$
   
   
   where $\vert x\vert<1$ or $n$ is a non negative integer and $x$ is any complex number.

5. Log series
   
   $$\log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} +\cdots + (-1)^... ...n+1} + \cdots \mbox{ where $x$ is real with $\vert x\vert<1$}.$$
   
   

6. Arctan series
   
   $$\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} + \cdots + (-1)^n\frac{x^{2n+1}}{2n+1} + \cdots$$
   
   
   where $x$ is real with $\vert x\vert<1$.

7. Arcsine series
   
   $$\arcsin(x) = x+\frac{1}{2}\frac{x^3}{3} +\frac{(1)(3)}{(2)(4)}\frac{x^5}{5} +\frac{(1)(3)(5)}{(2)(4)(6)}\frac{x^7}{7}+\cdots$$
   
   
   where $x$ is real with $\vert x\vert<1$.