Golden Ratio Series – Comic Book Inspired Sequence

Question

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?

Answer 1

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, $$ \begin{align} \frac{13}8+\sum_{n=0}^\infty\frac{(-1)^{n+1}(2n+1)!}{(n+2)!\,n!4^{2n+3}} &=\frac{13}8+4f\left(\frac1{16}\right)\\ &=\frac{13}8+4\left(\frac14\left(1+\frac14\right)^{1/2}-\frac1{32}-\frac14\right)\\ &=\frac{1+\sqrt5}2\\[8pt] &=\phi\tag{3} \end{align} $$