If series $\sum a_n$ converges and $\sum a_n^{2}$ diverges, prove that $\sum a_n$ converges under certain conditions and give an example of such series.
Any idea on how to prove this?
I tried to use tests for convergence, to possibly get closer to result to see what $a_n$ should be. But I am completely lost.
[Math] Square of convergent series
sequences-and-series
Best Answer
HINT: If $\sum_na_n$ converges, you know that $\lim_{n\to\infty}a_n=0$. This means that for all sufficiently large $n$, $|a_n|<1$, and therefore $a_n^2<|a_n|$. Thus, if the terms $a_n$ were positive, $\sum_na_n^2$ would have to converge even faster than $\sum_na_n$. But we’re told that $\sum_na_n^2$ diverges. This means that the $a_n$ cannot all be positive. They can’t even all be positive for all sufficiently large $n$. Nor can they all be negative from some point on. (Why?) What kind of convergent series must $\sum_na_n$ therefore be? Once you see that, it’s not hard to come up with an example.