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What requirements are needed on $a_i$, a non-negative sequence to ensure that $\sum^{\infty}_{i=0}a_i^2 < \infty$?

My answer:

If the series converges then it must be that $a_i^2$ approaches $0$, so $a_i$ also approaches $0$. So, $a_i$ approaches $0$ and must be bounded.

But, being bounded and $a_i$ approaching $0$ does not guarantee that the series will converge. For example, take $a_i = 1/\sqrt{i}$, then $a_i$ approaches $0$ and its bounded, but the series will diverge since $a_i^2 = 1/i$.

Question: What condition should be imposed then on $a_i$ to ensure that the series of $a_i^2$ will always diverge.? Is there any iff statement that would be applicable?

Jennie Shar
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    There is no “perfect” test: https://math.stackexchange.com/q/1174547/42969 – Martin R Sep 13 '24 at 14:35
  • Note that with $b_n = a_n^2$ your question is equivalent to asking under which conditions a series $\sum b_n$ of nonnegative terms converges. – Martin R Sep 13 '24 at 14:36
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    See also https://math.stackexchange.com/a/452074/42969: For every converging series there is another series with “much bigger terms” which still converges, and for every diverging series there is another series with “much smaller terms” which still diverges. – Martin R Sep 13 '24 at 14:38
  • @MartinR Thank you for the link to the other post and the comment. After reading it over, I would like to clarify. My question is the same as asking what conditions on $b_n$ would the series of $b_n$ converge. But, there is no perfect test, so there is no direct answer for that? It will converge if the sequence of partial sums are bounded, and that is as precise as it can get? What if $b_n$ is decreasing to $0$ and a bounded sequence. Then, is there a condition that can be imposed on the rate at which $b_n$ decreases to get convergence of the series? (Or, this also not something we can define) – Jennie Shar Sep 13 '24 at 16:28
  • There is no simple necessary and sufficient condition that is not essentially a restatement of "the sum converges". – Robert Israel Sep 13 '24 at 16:33

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