What is the sum $\sum_{k=0}^{\infty} kz^{-k}$?

Question

Any hint clarifying the problem as stated in the title, i.e

what is $\sum_{k=0}^{\infty} kz^{-k}$?

would be very appreciated.

Answer 1

Hint: Letting $w=\frac{1}{z}$ this is the same as: $$\sum_{k=0}^{\infty} kw^k.$$

This is a slightly more well-known series.

This will converge when $|w|<1$ and hence when $|z|>1.$

Answer 2

Hint:

Consider the entire function of $z^{-1}$

$$f(z^{-1})=\sum_{k=0}^\infty(z^{-1})^k,$$

which converges to $$\frac1{1-z^{-1}}$$ for all $|z|>1$.

Now by termwise differentiation,

$$f'(z^{-1})=\sum_{k=0}^\infty k(z^{-1})^{k-1}=z\sum_{k=0}^\infty k(z^{-1})^k.$$