Finding Region of Convergence

Question

I am trying to find the regions of absolute and uniform convergence for three different series, but I figured I'd start with the simplest one $$\sum_{k\geq 2} k(k-1)z^{k-2}$$

I have worked with Weiserstraß to prove that series converge over different given regions, but I've never been asked to find the region before.

Just based on Calc II type stuff I'm feeling like $|z|<1$ will give absolute convergence, but again have no idea how to show that formally in complex, and some of the subsequent series don't seem so simple.

Thanks

Answer 1

Here's two approaches to find the region of convergence:

1)

Write the sum as $\displaystyle{\sum_{k=0}^\infty} (k+2)(k+1)z^k$. Then, the radius of convergence will be $\lim \sup \sqrt[n]{k(k-1)} = 1$. Therefore, the series converges when $|z|<1$.

2)

Consider f(z) = $\displaystyle{\sum_{n=0}^\infty} z^n$. You may know that this converges when $|z|<1$ (it's the infinite geometric series). Deriving twice, we've got that $f''(z) = \sum_{k\geq 2} k(k-1)z^{k-2}$. Now, since deriving wrt $z$ a power series doesn't change the radius of convergence, it will be 1, so your series converges when $|z|<1$.