I'm hoping someone might be able to verify my solution to the following problem:
Suppose that the series $\sum c_n z^n$ has radius of convergence $R$. Find the radius of convergence of the $\sum n^p c_n z^n$.
My solution: Let $(a_n)$ be the sequence of coefficients for the series. We have $(a_n) = (n^pc_n)_n$. Using the ratio test,
$$ L = \lim_{n\to\infty} \frac{a_n}{a_{n+1}} = \lim_{n \to \infty} \frac{n^p c_n}{(n+1)^pc_{n+1}} = \lim_{n\to\infty} \frac{n^p}{(n+1)^p} \cdot \lim_{n\to\infty} \frac{c_n}{c_{n+1}} = R \cdot \lim_{n\to\infty} \left(\frac{n}{n+1}\right)^p = 1. $$
Thus, the radius of convergence is $1$.
$$ \limsup_{n\to\infty}\sqrt[n]{n^p|c_n|}=\left(\lim_{n\to\infty}\sqrt[n]{n}\right)^p\limsup_{n\to\infty}\sqrt[n]{|c_n|}=1\cdot\frac{1}{R}, $$ hence the radius of convergence is $R$.