Romanov's Series

Calculus Level 5

At first, I was amazed to those who could answer my recent OPs. I wonder, what kind of approach did they use to solve my OPs like:

  1. Possibly the strangest calculus problem??

  2. Possibly the strangest calculus problem?? (Part 2)

  3. Romanov's Integral

  4. Triple Cotangents Integral

But then I have a suspicion that they probably cheated since they answered them but posted no solutions. Besides, those problems can't be tackled by using softwares like Wolfram|Alpha, Mathematica, Maple, etc unless using special algorithms. Well, I hope I am wrong. (◠‿◠)

Anyway, here is a problem that I've just created.

\[ \begin{equation} \large\sum_{k=1}^\infty (-1)^{k}\frac{2k-1}{(2k-1)^4-16}=\frac{\color{red}{a}\pi+\color{red}{b}\pi\,{\rm{sech}}\,\color{red}{c}\pi}{\color{red}{d}} \end{equation} \]

Compute \(\large\, \color{red}{a}+\color{red}{b}+\color{red}{c}+\color{red}{d}\).


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