If for $a_n \geq 0$, the mean $\frac{1}{N}\sum_{n=1}^{N}a_n$ converges, then does $\lim_{n \rightarrow \infty} \frac{a_n}{n} = 0$?


1 Answer 1


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 $n\to\infty$.

  • $\begingroup$ 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. $\endgroup$ Dec 6, 2017 at 0:31
  • $\begingroup$ @AlfredYerger: Jack's example is wrong. This is a simple derivation. It is easy to judge whether it is correct or not. $\endgroup$
    – Hans
    Dec 6, 2017 at 0:34
  • $\begingroup$ 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. $\endgroup$ Dec 6, 2017 at 0:35
  • $\begingroup$ Jack you are mistaken. $\endgroup$ Dec 6, 2017 at 0:37
  • 1
    $\begingroup$ @uniquesolution: you are right guys, sorry for the mess and (+1) to Hans. $\endgroup$ Dec 6, 2017 at 0:38

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