<s>[INST] Calculate $\sum\limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$ Calculate $$\sum \limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$$
I use software to complete the  series is  $\frac{2}{27} \left(18+\sqrt{3} \pi \right)$
I have no idea about it.  :|
 [/INST] Consider the function
$$f(x) = \frac{\arcsin{x}}{\sqrt{1-x^2}}$$
$f(x)$ has a Maclurin expansion as follows:
$$f(x) = \sum_{n=0}^{\infty} \frac{2^{2 n}}{\displaystyle (2 n+1) \binom{2 n}{n}} x^{2 n+1}$$
Differentiating, we get
$$f'(x) = \frac{x \, \arcsin{x}}{(1-x^2)^{3/2}} + \frac{1}{1-x^2} = \sum_{n=0}^{\infty} \frac{2^{2 n}}{\displaystyle \binom{2 n}{n}} x^{2 n}$$
Evaluate at $x=1/2$
$$\sum_{n=0}^{\infty} \frac{1}{\displaystyle \binom{2 n}{n}} = \frac{\frac12 \arcsin{\frac12}}{3 \sqrt{3}/8} + \frac{4}{3} = \frac{2\sqrt{3} \pi+36}{27}$$
ADDENDUM
There are many derivations here of the above result for the Maclurin series for $f(x)$; I refer you to this one.
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