How one evaluates a particular series?

Question

How do one compute $\sum_{i=0}^\infty \frac{(2i+1)^5}{1+e^{(2i+1)\pi}}$? Wolfram alpha gives $31/504$ but what is the method to evaluate the series?

Answer 1

Using Bernoulli numbers $$S_n=\sum_{i=0}^\infty \frac{(2i+1)^n}{1+e^{(2i+1)\pi}}$$ $$S_{4p+1}=\frac{2^{4 p+1}-1}{4(2 p+1)}\,B_{2 (2 p+1)}$$ which generates the sequence $$\left\{\frac{1}{24},\frac{31}{504},\frac{511}{264},\frac{8191}{24},\frac{5749691557 }{28728},\frac{162912981133}{552},\frac{22076500342261}{24},\cdots\right\}$$

Using the zeta function, we can also write $$S_{4p+1}=\left(\frac{1}{2}-2^{-(4 p+2)} \right)\Gamma (4 p+2)\frac{\zeta(4p+2) }{\pi^{4p+2} }$$