AP Calculus BC Series Formulas

 $\sum_{n=0}^{\infty}c_n\left(x-a\right)^n=c_0+c_1\left(x-a\right)+c_2\left(x-a\right)^2+...+c_n\left(x-a\right)^n+...$  

• Power series centered at  $x=a$  

• Derivative of a power series centered at  $x=a$  

• Integral of a power series centered at  $x=a$  

• Geometric series

 $\sum_{n=1}^{\infty}nc_n\left(x-a\right)^{n-1}=c_1+2c_2\left(x-a\right)+...+nc_n\left(x-a\right)^{n-1}+...$  

• Power series centered at $x=a$  

• Derivative of a power series centered at  $x=a$  

• Integral of a power series centered at  $x=a$  

• Geometric series

 $\sum_{n=0}^{\infty}\frac{c_n\left(x-a\right)^{n+1}}{n+1}=c_0\left(x-a\right)+\frac{c_1\left(x-a\right)^2}{2}+\frac{c_2\left(x-a\right)^3}{3}+...+\frac{c_n\left(x-a\right)^{n+1}}{n+1}+...$  

• Power series centered at x = a

• Derivative of a power series centered at x = a

• Integral of a power series centered at x = a

• Geometric series

 $\sum_{n=1}^{\infty}ar^{n-1}=a+ar+ar^2+...+ar^n+...,\ \ \left|r\right|<1$  

• Power series centered at  $x=a$  

•  $\ln\left(1+r\right)$  
  

• Taylor Series

• Convergent Geometric series =  $\frac{a}{1-r}$  

 $\sum_{n=0}^{\infty}x^n=1+x+x^2+...+x^n+...,\ \ \left|x\right|<1$  

• Power series centered at x = 0

•  $\ln\left(1+x\right)$  
  

•  $\frac{1}{1-x}$  
  

•   $\frac{1}{1+x}$  

Tags

CCSS.HSF.BF.A.2

 $\sum_{n=0}^{\infty}\left(-1\right)^nx^n=1-x+x^2-...+\left(-x\right)^n+...,\ \ \left|x\right|<1$  

•  $e^x$  
  

•  $\ln\left(1+x\right)$  
  

•  $\frac{1}{1-x}$  
  

•   $\frac{1}{1+x}$  

 $\sum_{n=0}^{\infty}\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}+...,\ \ all\ real\ x$  

•  $e^x$  
  

•  $\ln\left(1+x\right)$  
  

•  $\cos\left(x\right)$  
  

•   $\sin\left(x\right)$  

 $\sum_{n=0}^{\infty}\frac{\left(-1\right)^nx^{2n+1}}{\left(2n+1\right)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}+...+\frac{\left(-1\right)^nx^{2n+1}}{\left(2n+1\right)!}+...,\ \ all\ real\ x$  

•  $e^x$  
  

•  $\tan^{-1}\left(x\right)$  
  

•  $\cos\left(x\right)$  
  

•   $\sin\left(x\right)$  

 $\sum_{n=0}^{\infty}\frac{\left(-1\right)^nx^{2n}}{\left(2n\right)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}...+\frac{\left(-1\right)^nx^{2n}}{\left(2n\right)!}+...,\ \ all\ real\ x$  

•  $e^x$  
  

•  $\ln\left(1+x\right)$  
  

•  $\cos\left(x\right)$  
  

•   $\sin\left(x\right)$  

 $\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n-1}x^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}+...+\frac{\left(-1\right)^{n-1}x^n}{n}+...,\ \ -1<x\le1$  

•  $e^x$  
  

•  $\ln\left(1+x\right)$  
  

•  $\cos\left(x\right)$  
  

•   $\sin\left(x\right)$