The method of telescoping continued fractions

We give an approach to discover continued fractions for series of the form $$\sum_{k=0}^\infty \frac{ε^k}{(x+k)^s},$$ where $ε= \pm 1$. We find a continued fraction of the form \begin{equation*} \frac{a_1}{b_1(x)} \fplus \frac{a_2}{b_2(x)} \fplus \fdots \end{equation*} where $a_k$ are constants an