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If $a_n$ is a sequence of non-increasing positive numbers, then suppose we already know that $$\sum_p a_p$$ converges, when $p$ runs over the primes, what should be used to prove that $$\sum_n \frac{a_n}{\log{n}}$$ also converges, where $n$ runs over the positive naturals?

And also, how to show the converse is also true?

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    Do you mean that the first series is summed over all primes and the second series is summed over all positive integers?2011-10-16
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    Sounds intuitively plausible, given the Prime Number Theorem.2011-10-16
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    To me non-increasing would mean that the sequence is not an increasing sequence - here I guess it should be a monotone decreasing sequence that is $a_{n+1}\le a_n$.2011-10-16
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    @AD: $a_{n+1}\le a_n$ *is* [non-increasing](http://en.wikipedia.org/wiki/Sequence#Types_and_properties_of_sequences). Monotone decreasing would be $a_{n+1}. I have seen in some places $(-1)^n$ being called a non-increasing sequence, but that is not the standard mathematical usage, since it does repeatedly increase from $-1$ to $1$.2011-10-16
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    @robjohn: I call that *strictly decreasing* - see http://en.wikipedia.org/wiki/Monotonic_function2011-10-16
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    @AD: strictly decreasing and non-increasing are the terms used to disambiguate decreasing which could mean either.2011-10-16
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    @robjohn: You just said that *monotone decreasing* adopts a *strict* inequality.2011-10-16
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    @AD: there are various usages. Just saying "decreasing" is usually ambiguous. Monotone decreasing is less ambiguous and usually means strictly decreasing, which is $a_{n+1}. non-increasing definitely means $a_{n+1}\le a_n$. I have also seen monotone non-increasing also.2011-10-16
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    @robjohn, what planet are you from? You have seen $(-1)^n$ being called a non-increasing sequence on Alpha Centauri B?2012-12-21
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    @TonyK: I don't remember exactly; it might have been on Yahoo answers (that's close to Alpha Centauri B, isn't it?), but I think the people were getting "non-increasing" mixed up with "not increasing".2012-12-21

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