On the alternating series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} \ln n}{n^2}$

Question

Does there exist a closed form for the series

$$\mathcal{S} = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} \ln n}{n^2}$$

Of course it is just $\eta'(2)$ where $\eta$ is the eta Dirichlet function. Wolfram Alpha does not return a result. On the other hand we are all aware of $\zeta'(2)$ which involves Glashier - Kinkelin constant.

How about this one?

$$\eta'(2) = 1/12 π^2 ( \gamma - 12 \log(A) + \log(4 π))$$ — Zaid Alyafeai, Apr 19 '17 at 09:07
Thanks Z !! However it is strange that this closed form is not mentioned nor in Wiki nor in Wolfram Wolrd! — Tolaso, Apr 19 '17 at 09:08
See here https://www.wolframalpha.com/input/?i=d%2Fds+DirichletEta(s)+,+s%3D2 — Zaid Alyafeai, Apr 19 '17 at 09:09
That might be useful as well https://math.stackexchange.com/questions/2210129/how-can-we-prove-that-2e2-int-0-infty-x-ln-x-over-1-e2e-pix-mathrm/2210655#2210655. — Zaid Alyafeai, Apr 19 '17 at 09:11

score 3 · Accepted Answer · answered Apr 19 '17 at 08:57

3

If $$\eta(s)=1-\frac1{2^s}+\frac1{3^s}-\frac1{4^s}+\cdots$$ then $$\eta(s)=\zeta(s)\left(1-\frac{2}{2^s}\right).$$ Therefore $\eta'(2)$ can be written explicitly in terms of $\zeta'(2)$.

answered Apr 19 '17 at 08:57

Angina Seng

158,341

On the alternating series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} \ln n}{n^2}$

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