How do you find the derivative of a power series?

Question

legomyego180 · Answer

So I have the series:
$$\sum_{}^{}\frac{ x^k }{ k^23^k }$$
I've solved for the Interval and Radius of convergence which I got to be $r=3, [-3,3]$

agent0smith · Answer

I still haven't gone to in n out, but wasn't there some about that on the link i posted in the other question?

legomyego180 · Answer

didnt see anything

zepdrix · Answer

Ummm well differentiation is linear operation... I think...
like in the sense that$$\large\rm \frac{d}{dx}(f+g)=\frac{d}{dx}f+\frac{d}{dx}g$$So if you have some big sum...
The derivative of a sum should be the sum of derivatives,$$\large\rm \frac{d}{dx}\sum \frac{1}{k^2 3^k}x^k\quad=\quad \sum \frac{1}{k^2 3^k}\frac{d}{dx}x^k$$Ya? I think? :d

legomyego180 · Answer

yea, agent was kind of explaining that earlier

legomyego180 · Answer

Ah I think im gonna crash, it's pretty late here. Ill leave the question open in case you guys find any helpful information or think of something. Thanks a ton for all of the help today @zepdrix @agent0smith

phi · Answer

you can take the derivative term by term to get a new series. You may have to tweak the bounds