series convergence on limit

Question

let (an) be as sequence of reals such that lim(n^2*an) exists in R.Prove or disprove that series (an) converges i know that as limit is given to be finite real number hence sequence (an) must be sum 1/infinity form but am not being able to do this. please help

Answer 1

If the sequence $(a_n)$ is such that $\lim\limits_{n \to \infty} n^2 a_n$ exists, then the sequence $(n^2 a_n)$ is eventually bounded (let's say for $n \ge N$) by $1$.

Then for $p>n >N$:

$$\left\vert\sum_{k=n}^p a_n\right\vert \le \sum_{k=n}^p \vert a_n \vert\le \sum_{k=n}^p \frac{1}{n^2}$$

Proving that the series $(a_n)$ converges as $\sum \frac{1}{n^2}$ converges.