How to compute this series: $$\sum_{k=0}^\infty \frac{C_k}{2^{2k+1}}$$ where $C_k$ is the catalan number: $C_k=\frac{1}{k+1}{2k \choose k}$. (Further, is there any general method to treat this question with general $C_k$?)

$\begingroup$ Use the power series for $\arcsin$. Not appropriate for MO. $\endgroup$– abxCommented Dec 31, 2018 at 18:21

$\begingroup$ The Taylor series for $\arcsin$ together with the binomial formula for $C_n$ does the trick, however, recognizing $\arcsin$ is not trivial at all. I therefore vote to reopen. $\endgroup$– JanChristoph SchlagePuchtaCommented Jan 2, 2019 at 9:08
4 Answers
The ordinary generating function of the $C_k$ is wellknown:
$$\sum_{k \geq 0} C_k x^k = \frac{1  \sqrt{1  4x}}{2x}$$
This can be deduced from the problem of counting binary planar trees and using familiar generatingfunctionology techniques. From there the problem is not difficult.
$k$th summand is the probability that a symmetric random walk with steps $\pm 1$ returns first time to the initial point at time $2(k+1)$. Since this random walk is recurrent, the sum of probabilities equals 1.

1$\begingroup$ @mathworker21 There are many independent proofs of recurrence, which work in more general setting. $\endgroup$ Commented Jan 1, 2019 at 8:02
And as it often happens when the sum of a series is rational, it is simply telescopic: $$ \frac{C_k}{2^{2k+1}}=\frac{\binom{2k}k}{4^k}\frac{\binom{2 (k+1) }{k+1} }{4^{k+1}}. $$
Maple 2018.2.1 and Mathematica 11.3.0 are your friends
sum(binomial(2*k, k)/((k+1)*2^(2*k+1)), k = 0 .. infinity);
$1$
Sum[Binomial[2*k, k]/(k + 1)/(2^(2*k + 1)), {k, 0, Infinity}]
$1$
sum(binomial(2*k, k)*x^k/(k+1), k = 0 .. infinity, parametric);
$$\cases{2\, \left( 1+\sqrt {14\,x} \right) ^{1}&$4\, \left x \right \leq 1$\cr \infty &$1<4\,x$\cr \sum _{k=0}^{\infty }{\frac {{2\,k\choose k}{x}^{k}}{k+1}}&otherwise\cr} $$
Sum[Binomial[2*k, k]/(k + 1)*x^k, {k, 0, Infinity},GenerateConditions > True]
$$\text{ConditionalExpression}\left[\frac{1\sqrt{14 x}}{2 x},x=\frac{1}{4}\lor x=\frac{1}{4}\lor \left x\right <\frac{1}{4}\right] $$
Edit. MMA code is added.