<s>[INST] A golden ratio series from a comic book The eighth installment of the Filipino comic series Kikomachine Komix features a peculiar series for the golden ratio in its cover:

That is,
$$\phi=\frac{13}{8}+\sum_{n=0}^\infty \frac{(-1)^{n+1}(2n+1)!}{(n+2)!n!4^{2n+3}}$$
How might this be proven?
 [/INST] Using the series
$$
(1-4x)^{-1/2}=\sum_{n=0}^\infty\binom{2n}{n}x^n\tag{1}
$$
we get
$$
\begin{align}
f(x)
&=\sum_{n=0}^\infty\frac{(-1)^{n+1}(2n+1)!}{(n+2)!\,n!}x^{n+2}\\
&=\frac12\sum_{n=1}^\infty(-1)^n\binom{2n}{n}\frac{x^{n+1}}{n+1}\\
&=\frac12\int_0^x\left[(1+4t)^{-1/2}-1\right]\,\mathrm{d}t\\
&=\frac14(1+4x)^{1/2}-\frac x2-\frac14\tag{2}
\end{align}
$$
Therefore,
$$
\begin{align}
\frac{13}8+\sum_{n=0}^\infty\frac{(-1)^{n+1}(2n+1)!}{(n+2)!\,n!4^{2n+3}}
&=\frac{13}8+4f\left(\frac1{16}\right)\\
&=\frac{13}8+4\left(\frac14\left(1+\frac14\right)^{1/2}-\frac1{32}-\frac14\right)\\
&=\frac{1+\sqrt5}2\\[8pt]
&=\phi\tag{3}
\end{align}
$$
</s>