help please?

ilovecalculus · February 27th 2006, 07:55 PM

hey- how do i find a power series for f'(x) if

f(x)= sum from n=0 to n=infinity [(-1^n)*x^(2n)]/[(2n)!]

thanks.

**CaptainBlack** · February 27th 2006, 08:58 PM

Originally Posted by ilovecalculus

hey- how do i find a power series for f'(x) if

f(x)= sum from n=0 to n=infinity [(-1^n)*x^(2n)]/[(2n)!]

thanks.

How to find a power series representation of $f'(x)$ if:

$ f(x)=\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} $ ?

Differentiate term by term (I know this works for this series so I won't worry
about convergence).

$ f'(x)=\sum_{n=0}^{\infty} (-1)^n \frac{\frac{d(x^{2n})}{dx}}{(2n)!} $ ,

so:

$ f'(x)=\sum_{n=0}^{\infty} (-1)^n \frac{2n\ x^{2n-1}}{(2n)!} $ ,

Simplifying slightly:

$ f'(x)=\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n-1}}{(2n-1)!} $ .

RonL

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help please?