Common Series

• Geometric
  
  $$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots (-1 < x < 1 )$$
  
  

• Binomial
  
  $$(1+x)^n = 1 + nx + \frac{n(n-1)}{2} x^2 + + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots (-1 < x < 1)$$
  
  

• Logarithm
  
  $$\ell n\,(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots (-1 < x < 1)$$
  
  

• Exponential
  
  $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \,;\, (-\infty \le x \le +\infty )$$
  
  

• Sine
  
  $$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!}- \cdots \,;\, (-\infty \le x \le +\infty )$$
  
  

• Cosine
  
  $$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \,;\, (-\infty \le x \le +\infty )$$