• February 27th 2006, 07:55 PM
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.
• February 27th 2006, 08:58 PM
CaptainBlack
Quote:

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

$
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