# Finding the radius of convergence for $\sum n^p z^n$ (Proof Verification)

Goal: Find the radius of convergence for the following complex power series:

$$\sum n^p z^n$$

Attempt:

1. We have by Hadamard's formula for the radius of convergence that the complex power series $\sum a_n z^n$ converges if $|z| < R$ s.t.

$$R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} |a_n|^{1/n}}$$

2. Then we here have that $a_n = n^p$ for all $n \in \mathbb{N}$.

3. Then applying Hadamard's formula we obtain that

$$R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} |n^p|^{1/n}}$$

4. Then since

$$\underset{n \rightarrow \infty}{\limsup} |n^p|^{1/n} = 1$$

no matter the value of $p$, we have that $R = 1$.

Is this correct?

$$1 > \lim\frac {|(n+1)^pz^{n+1}|} {|n^pz^p|} = \lim \left(1 + \frac 1n\right)^p |z| = |z|$$ and divergence as soon as $$1 < \lim\frac {|(n+1)^pz^{n+1}|} {|n^pz^p|} = |z|$$ so $R = 1$.