How would one go about computing the radius of convergence of, say, the following power series:
$$\sum_{n=-\infty}^0 n \, 3^{-n} z^n.$$
It is tempting to directly apply the Cauchy-Hadamard theorem here, but the statement is true for power series summing from $n=0$. I tried to make a substitution by realizing the sum above is equal to :
$$\lim_{m \to \infty} \sum_{n=-m}^0 n \, 3^{-n} z^n,$$
and letting $k = n + m$ for $n= -m, -m+1, \ldots, 0$ we have
$$\lim_{m \to \infty} \sum_{k=0}^m (k-m) \, 3^{m-k} z^{k-m}.$$
But this seems like making things worse. Any thoughts on tackling this problem?
$$\sum_{n=0}^{\infty}(-n)3^nz^{-n}$$ converges for $|z|>3$.