TITLE: Radius of convergence of $\sum\limits_{n \ge 1} a_n z^n$ where $a_n$ is the number of divisors of $n^{50}$
QUESTION [6 upvotes]: Consider the power series $\sum_{n \ge 1} a_n z^n$, where $a_n$ is the number of divisors of $n^{50}$. What is its radius of convergence?

My attempt
$a_n < n^{50}$ $\forall$ $n$.
So $\lim \{a_n\}^{\frac{1}{n}} \le \{\lim n^{\frac{1}{n}}\}^{50} = 1$.
So radius of convergence is 1.

REPLY [7 votes]: Hint: For every $n$, the number of divisors of $n^{50}$ is between $1$ and $n^{50}$. Determine the radiuses of convergence of the series $\sum\limits_nz^n$ and $\sum\limits_nn^{50}z^n$. Conclude.
One sees that the proof uses neither the number of divisors of $n^{50}$, nor its exact growth, nor any other number theoretical refinement.