<s>[INST] Convergence of sum of reciprocals of binomial coefficients I am interested in evaluating the sums
$$
\sum_{k=1}^{\infty}{\binom {2k}{k}^{-n}},
$$
where $n$ is a positive integer. It is already known that for $n=1$ we have
$$
\sum_{k=1}^{\infty}\frac{1}{\binom {2k}{k}}=\frac{9+2\sqrt 3 \pi}{27}.
$$
Several papers analyze the properties of sums involving the above identity (such as this one), however I was not able to find any material relating to the cases $n>1$. I already know these sums converge for all positive integers $n>0$, however I would be interested in finding a nice closed form for them as in the case $n=1$. How would I go about this? Are there results available about such sums in literature?
 [/INST] I think that, for $n>1$, you are entering in the world of hypergeometric functions. 
Just have a look to the table and notice the patterns
$$\left(
\begin{array}{cc}
n & S_n \\
 2 & \frac{1}{4} \, _3F_2\left(1,2,2;\frac{3}{2},\frac{3}{2};\frac{1}{16}\right) \\
 3 & \frac{1}{8} \,
   _4F_3\left(1,2,2,2;\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{1}{64}\right) \\
 4 & \frac{1}{16} \,
   _5F_4\left(1,2,2,2,2;\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{1}{25
   6}\right) \\
 5 & \frac{1}{32} \,
   _6F_5\left(1,2,2,2,2,2;\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{
   2};\frac{1}{1024}\right)
\end{array}
\right)$$ What is interesting is that $\log(S_n)$ is almost a linear function of $n$ (almost $\log(S_n)=-n \log(2)$).
</s>