I'm learning complex series and I have a doubt.
I have to find the radius of convergence and the sum of $$\sum_{2}^{\infty} 3nz^{n-1}$$
For $z=0$ I believe the series converges and the sum is $0$.
For $z\neq0$ I've tried to apply the ratio test but with no success.
This is what I have done.
$$\lim_{n \to \infty} \sqrt[n]{|3n\frac{z^n}{z}|}\Rightarrow \lim_{n \to \infty} |z|\sqrt[n]{\frac{3n}{z}} $$
Getting here I've realized that I've must have done something wrong.
Can someone point me in the right direction?
Best Answer
Try the quotient test (many times more gentle than the $\,n$-th root one):
$$\left|\;\frac{3(n+1)z^n}{3nz^{n-1}}\;\right|=\frac{n+1}n|z|\xrightarrow[n\to\infty]{}|z|<1$$