Q:

Finding radius of convergence where the coefficient, $a_n$ is the number of divisors of $n^{50}$, $\forall n$

Consider the power series $\sum_{n\ge1}a_nx^n$ where $a_n=$ number of divisors of $n^{50}$. What is the radius of convergence of this power series. 
The radius of convergence of  $\sum_{n\ge1}nx^n$ is 1, so I guess this would be $1$ too. But I can't derive it rigorously.

A:

As $1\le d(n^{50})\le n^{50}$, the radius of convergence
of $\sum_n d(n^{50})x^n$ is between that of $\sum_n x^n$ and $\sum_n n^{50}x^n$. These series both have radius of convergence $1$.

