Elementary summations of non-elementary functions The other day, while I was doing a bit of research into the Riemann Zeta function, I discovered a few interesting closed-form values of summations involving it, like
$$\sum_{s=2}^\infty (\zeta(s)-1)=1$$
$$\sum_{s=1}^\infty (\zeta(2s)-1)=\frac{3}{4}$$
and when I checked Wikipedia, I found a ton of other more complicated sums involving the Zeta Function, like
$$\sum_{s=2}^\infty \frac{\zeta(s)-1}{s}=1-\gamma$$
I know the trick that can be used for these sums is switching the order of the sums and using the fact that a geometric series is formed, but it fascinated me nevertheless.
QUESTION: Does anybody know of any other interesting examples of sums (that don't telescope trivially) with each term using non-elementary functions but a final answer that is elementary, or expressible in terms of other widely-known mathematical constants (like $\pi$, $e$, or $\gamma$)?
Specifically, I am looking for interesting identities regarding:


*

*sums involving the inverses of functions that don't have elementary inverses, like the Lambert-W function, or the inverse of the function $f(x)=x+\sin x$

*sums involving $\Gamma(s)$, at non-integer values of $s$ 

*interesting generating functions for sequences that currently have no discovered closed-form explicit formula, like the Bell, Bernoulli, or Harmonic numbers

*sums involving $\operatorname{Si}(s)$, $\operatorname{Li}(s)$, or $\operatorname{Ei}(s)$, or the elliptic integrals


Thanks!
 A: In the non-exhaustive list below, you can find a lot of series involving special functions.
Among them many are interesting. For example, this one involving $\Gamma$ and $\zeta$ :
$$\sum_{k=1}^\infty \frac{(-1)^k}{k!}\Gamma(k+x)\zeta(k+x)=-\Gamma(x)$$
Especially with $x=\frac{1}{2}$ one get a series representation for  $-\sqrt{\pi}$.
It should be too long to edit many others from :
http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/23/01/
http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/23/01/
http://functions.wolfram.com/Polynomials/HermiteH/23/02/
http://functions.wolfram.com/Polynomials/LaguerreL/23/02/
http://functions.wolfram.com/Polynomials/LegendreP/23/02/
http://functions.wolfram.com/Polynomials/ChebyshevT/23/01/
http://functions.wolfram.com/Polynomials/ChebyshevU/23/01/
http://functions.wolfram.com/Polynomials/GegenbauerC3/23/02/
http://functions.wolfram.com/Polynomials/JacobiP/23/01/
http://functions.wolfram.com/Polynomials/EulerE2/23/02/
http://functions.wolfram.com/Polynomials/BernoulliB2/23/02/
http://functions.wolfram.com/Polynomials/BellB2/23/02/
http://functions.wolfram.com/Polynomials/NorlundB2/23/02/
http://functions.wolfram.com/Polynomials/ZernikeR/23/01/
http://functions.wolfram.com/HypergeometricFunctions/ChebyshevTGeneral/23/01/
http://functions.wolfram.com/HypergeometricFunctions/GegenbauerC/23/01/
etc.
