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I'm having trouble with some theory..

Let's say I was given a positive Series as $\sum_{n=1}^\infty a_n$. Moreover, I know that $\lim_{n\to \infty} n^2a_n=\frac{1}{2}$.

At first I thought that because $\lim_{n\to \infty} n^2a_n=\lim_{n\to \infty} \frac{a_n}{\frac{1}{n^2}}=\frac{1}{2}$, then it reminds a lot the limit from Limit Comparison test with the Series $\frac{1}{n^2}$ which is convergent. So, can I tell from above-mentioned information, that $a_n$ converges as well?

On the other hand, I know that $lim_{n\to \infty}a_n$ must be 0 (otherwise the Series would diverge for sure), that I would be able to continue and test whether the given Series could be convergent. (but I have no info regarding $lim_{n\to \infty}a_n$)

So, from what it looks like, though the limit looks like Limit Comparison Test with convergent Series, yet I can't tell for sure it's ($\Sigma a_n$) a convergent Series.

Would appreciate your advice and example for $a_n$ as a diverging Series (in case there's) that fulfills the mentioned details.

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Let's define $b_n:=n^2a_n$. Then $\lim_{n\to\infty}b_n=1/2$; in particular, there is some $N\in\mathbb N$ such that $0\leq b_n\leq1$ for all $n\geq N$. Moreover, since $a_n=b_n/n^2$, we have $0\leq a_n\leq 1/n^2$ for all $n\geq N$. Now you can use the Comparison Test to conclude that $\sum_{n=1}^\infty a_n$ converges.

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$\sum_{n=1}^\infty a_n$ converges. You can use the limit comparison test with $a_n$ and $\frac 1{2n^2}$ As the ratio between them has a finite limit (here, $1$), the two sums either both converge or both diverge.

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