Finding the radius of convergence of the given power series.

Question

Consider the power series $\sum\limits_{n=1}^{\infty}$$a_nZ^n$ , where $a_n$ is the number of divisor of $n^{50}$ . Find the radius of convergence.

Answer 1

Use the facts that

1. $\tau(n) \le 2\sqrt{n}$ for all $n$
2. For infinitely many $n$ (say powers of $2$), $\tau(n) \ge \log_{2} n$
3. $n^{1/n} \to 1$ and $(\log n)^{1/n} \to 1$

Answer 2

We know that number of divisor of $n^{50}$ is $\tau$($n^{50}$) which tends to $\infty$ as n tends to $\infty$ . Let R be the radius of convergence, then $\frac{1}{R}$ = $\lim_{n\to \infty}$ sup $(a_n)^{\frac{1}{n}}$ is $\infty$, so R is 0. So please tell me am I right or not?