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I am reading this post. According to @ItsNotObvious when we have the series

$$\lim_{n \to \infty} \sum_{k=1}^n a_k = L $$ then $$\lim_{n \to \infty}a_n = 0$$

It is possible to impose other additional conditions to the sequence $a_n$ so as the above equations to hold for $a_n \geq 0$?

EDIT: Thanks to @WishofStar comment I found the following theorem

A series of nonnegative terms converges if and only if its partial sums form a bounded sequence (Rudin).

So, if we have that $a_n$ is a bonded sequence can we say that it's partial sums are also bounded and thus based on the theorem the terms $a_n$ has to be non-negative?

Thoth
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    If the series $\displaystyle\sum_{k=1}^na_k$ is monotonically increasing, then surely it is. – WishofStar Jan 26 '20 at 15:18
  • @WishofStar Is it possible to elaborate it a little more? I have little knowledge of real-analysis. – Thoth Jan 26 '20 at 15:21
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    What do you mean "come up with"? The $a_n$ could be positive, negative, or $0$. – lulu Jan 26 '20 at 15:22
  • I mean if there can be other conditions on $a_n$ so as to have $a_n \geq 0$ (I did a small edit.). – Thoth Jan 26 '20 at 15:26
  • Do you mean to ask if we can conclude that the terms $a_n$ are all non-negative, given conditions on the series? If so, the, @WishofStar has given you the answer. – Cameron Buie Jan 26 '20 at 16:41
  • Can we use @WishofStar statement when $L=0$? – Thoth Jan 26 '20 at 22:12

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