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

Question

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.

Answer 1

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 $1$.

Answer 2

For each $n\in\mathbb N$, $1\leqslant a_n\leqslant n^{50}$. Therefore:

• if $|x|\geqslant1$, the series diverges;
• if $|x|<1$, the series converges.

Therefore, the radius of convergence is $1$.