Given the following sum:
$$0.5\cdot\sum\limits_{k=0}^\infty \frac{1}{k+1}\binom{2k}{k}\cdot(0.25)^{k}$$ I know that the sum is supposed to converge to $1$. How would I go about evaluating it to get this result? I thought about binomial theorem, but the fraction $\frac{1}{k+1}$ makes it rather problematic.