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Power Series Representations of Functions

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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

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Power Series

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