How do you find the power series representation for the function $f(x)=sin(x^2)$ ?

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Answer 1 · 2014-10-06T23:22:57

Since

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

by replacing $x$ by $x^2$ ,

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

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

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