How can I find the value of $ \sum_{n = 1}^{\infty} A \cdot \frac{1 - e^{-Bn}}{(Cn + D)^3} $ in terms of the constants $ A $, $ B $, $ C $ and $ D $?

Question

I really don't know how to approach this. I plotted the sum on a graphing calculator and it definitely does converge - I would like to see how the limit depends on the constants.

Answer 1

$$\sum_{n = 1}^{\infty} A \frac{1 - e^{-Bn}}{(Cn + D)^3}=A\sum_{n = 1}^{\infty} \frac{1 }{(Cn + D)^3}-A\sum_{n = 1}^{\infty} \frac{e^{-Bn}}{(Cn + D)^3}$$ $$\sum_{n = 1}^{\infty} \frac{1 }{(Cn + D)^3}=-\frac{1}{2 C^3}\psi ^{(2)}\left(\frac{C+D}{C}\right)$$ $$\sum_{n = 1}^{\infty} \frac{e^{-Bn}}{(Cn + D)^3}=\frac{e^{-B}}{C^3}\,\,\Phi \left(e^{-B},3,\frac{C+D}{C}\right)$$

So you have your result in terms of the polygamma function and the Hurwitz-Lerch transcendent function.