Radius of convergence of $\sum\limits_{n \ge 1} a_n z^n$ where $a_n$ is the number of divisors of $n^{50}$

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.