<s>[INST] Series $\frac{1}{4}+\frac{1\cdot 3}{4\cdot 6}+\frac{1\cdot 3\cdot 5}{4\cdot 6\cdot 8}+\cdots$ Find the sum of the series to infinity$$\frac{1}{4}+\frac{1\cdot 3}{4\cdot 6}+\frac{1\cdot 3\cdot 5}{4\cdot 6\cdot 8}+\cdots$$
Attempt-
I wrote the general term as $$\frac{\binom{2n}{n}}{2^{2n}\cdot (n+1)}$$
I don't know what to do next
 [/INST] The numerator of the $n$-th term is
$$\frac{(2n)!}{2^nn!}\;,$$
and the denominator is
$$\frac{2^{n+1}(n+1)!}2=2^n(n+1)!\;,$$
so the term is actually
$$\frac{(2n)!}{2^{2n}n!(n+1)!}=\frac{1}{n+1}\binom{2n}{n}\left(\frac{1}{4}\right)^n\;.$$
Do you know anything about the Catalan numbers and generating functions?
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