Is there a nice (non-summation) representation of the following sum?
$$\sum_{n=1}^{N}C_{n}b^n$$ where $b <1$ and $C_{n}$ are the Catalan numbers. Obviously, this can be written as
$$\sum_{n=1}^{N}C_{n}b^n = \sum_{n=1}^{N}\binom{2n}{n}b^n -\sum_{n=1}^{N}\binom{2n}{n+1}b^n$$
I doubt there's a closed-form expression. We know that $$ \frac{1-\sqrt{1-4b}}{2}=\sum_{n=0}^{\infty}{C_nb^n} $$ is the generating function of the Catalan numbers, and you can try to bound the error with Taylor's formula.
Here the Catalan numbers are indexed starting at $1$, rather than at $0$ as is more common. I copied this formula from a paper that uses the unusual convention that $C_0=0, C_1=1, C_2=1, C_3=2, ...,$ so the lower index of summation might as well be $1$ as $0$.
Fortuitously, this formula would be easier to use in bounding the error with Taylor's formula than the more common generating function (with a $b$) in the denominator, because it would be easier to calculate the $n$th derivative.