this is a problem that's bugging me. It's a two part question. I believe I have answered part a. I'm not sure where to start for part b, please help. I will upload the question also the answer for part a. I like the way you have answered my question earlier that's why I'm coming back to you. Thank you for all your helpcenter image description here
2026-04-15 12:54:27.1776257667
Power series derivative
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Part a) is correct. The center of the power series is $2$ and its radius of convergence is $3$.
As regard part b), you should know that, for power series, in order to compute the derivative in the interval $(-1,5)$, you can take the derivative term by term (see the section Differentiation and integration here). Hence $$f'(x)=\sum_{n=1}^{\infty}\frac{n^2}{3^n}(x-2)^{n-1}=\sum_{n=0}^{\infty}\frac{(n+1)^2}{3^{n+1}}(x-2)^{n}$$ The radius of convergence of $f'$ is the same one of $f$, that is $3$ (check it!).