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 (t^{1/2}/2) dt

Thanks!!!

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- Nov 5th 2012, 05:31 PMkimsanders82Need 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 (t^{1/2}/2) dt

Thanks!!! - Nov 5th 2012, 07:10 PMProve ItRe: 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. - Nov 5th 2012, 07:11 PMTheEmptySetRe: Need help with power series representation of function
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

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

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