Sum $\sum\limits_{n=0}^{\infty}(-1)^{n-1}\frac1{(2n+1)^3}$

Asked Jun 26 '20 at 17:15

Active Jun 26 '20 at 17:39

Viewed 79 times

What is the value of the sum $$\sum\limits_{n=0}^{\infty}(-1)^{n-1}\frac1{(2n+1)^3}$$

I think the sum evaluates to $\frac{7\zeta(3)}{8}-\frac1{32}\zeta(3,-\frac1{4})$, where the last zeta is the Hurwitz zeta function. But, Hurwitz zeta is defined only for $\operatorname{Re}(q)>0$ as a Dirichlet series in this manner. Am I right? Any hints? Thanks beforehand.

edited Jun 26 '20 at 17:39

Aryaman Maithani

15,771

asked Jun 26 '20 at 17:15

vidyarthi

7,028

why the downvote? – vidyarthi Jun 26 '20 at 17:24
1

Did you mean instead for the $(1)^{n-1}$ to be in the summand? – RobPratt Jun 26 '20 at 17:24
1

Note: I have put the $(-1)^{n-1}$ within the summation. This is not what the OP had originally written but I assume is what they meant. – Aryaman Maithani Jun 26 '20 at 17:39

Sum $\sum\limits_{n=0}^{\infty}(-1)^{n-1}\frac1{(2n+1)^3}$

0 Answers0