# Telescoping series for $\zeta(s)$, question about the basic ideas and a specific series

There are many known telescoping series that enable analytic continuation of $\sum _n \frac {1}{n^{s}}$ into a variety of domains, however they seem to all be derived from two basic ideas:

1) The algebraic fact that $A-B+C-D+E-\dots=A+B+C+D+E+\dots-2\,(B+D+\dots)$ giving the Dirichlet $\eta$ series: $$\displaystyle \zeta(s) = \frac{1}{1-2^{1-s}}\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}, \qquad \Re(s) \gt 0$$

2) Subtracting an integral corresponding to a sum and then breaking the integral into integrals over subintervals like $\sum _n \frac{1}{n^{s}} - {1\over s-1} \;=\; \sum _n \Big({1\over n^s} - \int_n^{n+1}{dx\over x^s}\Big)$ giving:

$$\displaystyle \zeta(s) = \frac{1}{(s-1)} \left(1+\sum _{n=1}^{\infty } \left( {\frac {s-1-n}{n^{s}}} + \frac{n+1}{(n+1)^s}\right) \right), \qquad \Re(s) \gt 0$$

Q1: Are there any more of these basic ideas to derive telescoping series for $\zeta(s)$?

From these two basic versions, many different ones can be derived (examples here, here, here and here). I experimented quite a bit with combining the different versions and searched for the one with (imo) the highest 'symmetry' in the series component. In the end I found this one:

$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(s+\frac{s+1}{2^s}+\sum _{n=2}^{\infty } \left( {\frac {n+s}{(n+1)^{s}}} - \frac{n-s}{(n-1)^s}\right) \right), \qquad \Re(s) \gt -1$$

Q2: Is there any way to simplify these series (except for writing the denominator as $(n^2-1)^s$)?