# Thread: Need help with power series representation of function

1. ## Need help with power series representation of function

I need help with the following:
Find the power series representation of the function defined by:
F(x) = the integral from 0 to x of cos (t1/2/2) dt

Thanks!!!

2. ## Re: Need help with power series representation of function

By the second fundamental theorem of calculus, we have \displaystyle \displaystyle \begin{align*} F'(x) = \cos{\left( \frac{x^{\frac{1}{2}}}{2} \right)} \end{align*}.

If you find the power series for \displaystyle \displaystyle \begin{align*} F'(x) \end{align*} and integrate it, you will have the power series you need.

3. ## Re: Need help with power series representation of function

Originally Posted by kimsanders82
I need help with the following:
Find the power series representation of the function defined by:
F(x) = the integral from 0 to x of cos (t1/2/2) dt

Thanks!!!
We should know that the Taylor series for cosine is

$\displaystyle \cos(y)=\sum_{n=0}^{\infty}\frac{(-1)^{n}y^{2n}}{(2n)!}$

Now let $\displaystyle t=\frac{\sqrt{t}}{2}$

This gives

$\displaystyle \cos\left( \frac{\sqrt{t}}{2}\right)=\sum_{n=0}^{\infty}\frac {(-1)^{n}t^{n}}{4^n(2n)!}$

Now just integrate both sides from zero to x to get

$\displaystyle \int_{0}^{x}\cos\left( \frac{\sqrt{t}}{2}\right)dt=\int_{0}^{x}\sum_{n=0} ^{\infty}\frac{(-1)^{n}t^{n}}{4^n(2n)!} dt$

$\displaystyle F(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{n+1}}{4^n(2n)!(n+1)}$