How to calculate variant of geometric series based on sequences of Catalan numbers?

Question

I want to calculate

$$\sum_{n=0}^{\infty}\frac{(2n)!}{n!(n+1)!}k^{n+1}C$$

where $0<k<1$ and $C$ and $k$ are some constants, and $0<C$. Is there any possible range of $0<C$ that allows easy calculation of this?

In general, how do I solve this sum?

Answer 1

Hint: Differentiate with regard to the parameter k, and identify the binomial series behind the new expression.