If for $a_n \geq 0$, the mean $\frac{1}{N}\sum_{n=1}^{N}a_n$ converges, then does $\lim_{n \rightarrow \infty} \frac{a_n}{n} = 0$?
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Yes. Let $s_N=\frac{1}{N}\sum_{n=1}^{N}a_n$. Then $a_n=ns_n-(n-1)s_{n-1}$ and $$\frac{a_n}n=s_n-\Big(1-\frac1n\Big)s_{n-1}\to 0$$ as $n\to\infty$.
Hans
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This answer received quite a handful of up and down votes. Can we maybe discuss what we like or don't like about this? I thought it wasn't true and had an idea similar to Jack's. – A. Thomas Yerger Dec 06 '17 at 00:31
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@AlfredYerger: Jack's example is wrong. This is a simple derivation. It is easy to judge whether it is correct or not. – Hans Dec 06 '17 at 00:34
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Right, and the point of my question is to determine where the impasse is. Maybe that way some of the downvoters will switch them to upvotes. It appears Jack's answer was deleted anyway. I guess that also settles the issue. – A. Thomas Yerger Dec 06 '17 at 00:35
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Jack you are mistaken. – uniquesolution Dec 06 '17 at 00:37
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1@uniquesolution: you are right guys, sorry for the mess and (+1) to Hans. – Jack D'Aurizio Dec 06 '17 at 00:38
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Alfred, upvotes and downvotes are very dynamic, emotional, abrupt, spontaneous, and have little to do with mathematics, in general. – uniquesolution Dec 06 '17 at 00:39
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@Rajat: If you like my answer, please accept it. – Hans Dec 06 '17 at 18:47