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Does the series $$\sum_{n=2}^{\infty}\frac{\ln n}{n^2}$$ converge?

I'm searching for a solution that does not use the Integral test, Stirling, L’Hôpital or functions theorems.

I tried ratio test, and also comparing it to another series.

Thank you very much.

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    $\begingroup$ It is perhaps worth noting that the series is $-\zeta^' (2)$. $\endgroup$ – Eric Naslund Apr 21 '11 at 21:43
  • $\begingroup$ Please put your entire question in the body of your message. $\endgroup$ – Arturo Magidin Apr 21 '11 at 22:02
  • $\begingroup$ @Arturo:Ok, I will. $\endgroup$ – user6163 Apr 21 '11 at 22:21
  • $\begingroup$ Is there a particular reason why we're avoiding L'Hopital? It makes this vastly easier (in the sense that it's used often to prove david's answer below). $\endgroup$ – davidlowryduda Apr 21 '11 at 23:28
  • $\begingroup$ @mixedmath: Some analysis classes cover sequences/series before functions, maybe this problem comes from such a class.... $\endgroup$ – N. S. Nov 2 '11 at 23:14
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You can show that $\ln n\leq \sqrt n$ if $n$ is large enough. Now you can readily deduce that the series converges.

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You can use the condensation test for series whose terms are positive and weakly decreasing.

You replace the sum $$\sum_n a_n$$ by $$\sum_k 2^k a_{2^k}$$ and check that it converges.

Edited to add details:

$$\sum_{n=1}^{\infty}\frac{\ln n}{n^2} \longrightarrow \sum_{k=0}^{\infty}2^k \frac{\ln 2^k}{2^{2k}}=\sum_{k=0}^{\infty}\frac{k \ln 2}{2^k}$$

The last sum converges by the ratio test, or by identifying it as the derivative of a geometric series evaluated at $q=1/2$.

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  • $\begingroup$ I tried that before, It does not seem to work for me. $\endgroup$ – user6163 Apr 21 '11 at 21:17
  • $\begingroup$ $2^ka_{2^k}=\frac{k\ln 2}{4^k}$ and the ratio test shows that $\sum_k2^ka_{2^k}$ converges. $\endgroup$ – Davide Giraudo Apr 21 '11 at 21:22
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Here is a different style of answer, which may be useful to some.

$\zeta(s)=\sum_{n=1}^\infty n^{-s}$, the Riemann zeta function, is analytic on $\Re(s)>1$. This is because on any half plane $\Re(s)>1+\epsilon$, it is a uniformely convergent series of analytic functions. Its derivative, $$\zeta^{'}(s)=-\sum_{n=1}^\infty \log n n^{-s}$$ then also converges on all of $\Re(s)>1$. Now, notice that your sum is $-\zeta^{'}(2).$

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