Power Series Representations of Functions

Key Questions

• How do you find the power series representation for the function #f(x)=e^(x^2)# ?

  Since

  #e^x=sum_{n=0}^infty{x^n}/{n!}#,

  by replacing #x# by #x^2#,

  #e^{x^2}=sum_{n=0}^infty{(x^2)^n}/{n!}=sum_{n=0}^infty{x^{2n}}/{n!}#.

  I hope that this was helpful.

  Wataru · · Oct 5 2014
• How do you find the power series representation for the function #f(x)=(1+x)/(1-x)# ?

  Recall:
  #1/{1-x}=sum_{n=0}^infty x^n=1+sum_{n=1}^infty x^n#
  By multiplying by #x#,
  #x/{1-x}=sum_{n=0}^infty x^{n+1}=sum_{n=1}^infty x^n#

  So,
  #{1+x}/{1-x}=1/{1-x}+x/{1-x}=1+sum_{n=1}^infty x^n+sum_{n=1}^infty x^n#
  #=1+2sum_{n=1}^infty x^n#

  Wataru · · Sep 9 2014
• How do you find the power series representation for the function #f(x)=sin(x^2)# ?

  Since

  #sinx=sum_{n=0}^infty(-1)^n{x^{2n+1}}/{(2n+1)!}#,

  by replacing #x# by #x^2#,

  #Rightarrow f(x)=sum_{n=0}^infty(-1)^n{(x^2)^{2n+1}}/{(2n+1)!}#

  #=sum_{n=0}^infty(-1)^n{x^{4n+2}}/{(2n+1)!}#

  Wataru · 5 · Oct 6 2014

• How do you find the power series representation for the function #f(x)=ln(5-x)# ?
• How do you find the power series representation of a function?
• How do you find the power series representation for the function #f(x)=sin(x^2)# ?
• How do you find the power series representation for the function #f(x)=cos(2x)# ?
• How do you find the power series representation for the function #f(x)=e^(x^2)# ?
• How do you find the power series representation for the function #f(x)=tan^(-1)(x)# ?
• How do you find the power series representation for the function #f(x)=(1+x)/(1-x)# ?
• How do you find the power series representation for the function #f(x)=1/(1-x)# ?
• How do you find the power series representation for the function #f(x)=1/((1+x)^2)# ?
• How to find the Laurent series about #z=0# and therefore the residue at #z=0# of #f(z) = 1/(z^4 sin(pi z))#, where #f(z)# is a complex valued function?
• Question #87417