Calculus – Prove Divergence of $\sum n^2 a_n$ Given Conditional Convergence of $\sum a_n$

Question

Ratio and root tests won't help. And I can't use the comparison test because $ |a_n| $ is not necessarily smaller than $ n^2a_n $.

Can I use limits? We know:

1. $\lim\limits_{n \to \infty} a_n = 0 $
2. $\lim\limits_{n \to \infty} |a_n| \ne 0 $

And we need to prove:

1. $\lim\limits_{n \to \infty} n^2a_n \ne 0 $

Any ideas/hints?

Answer 1

Suppose that $n^2 a_n$ had a convergent series: That is,

$$\sum_{n = 1}^{\infty} n^2 a_n = \ell \in \mathbb{R}$$

Then $n^2 a_n \to 0$ as $n \to \infty$, so $n^2 a_n$ is eventually less than $1$ in absolute value, implying that $$|a_n| < \frac 1 {n^2}$$

for sufficiently large $n$. Now think about the comparison test.