Proving a formula for $\pi$

Question

I found a formula for $\pi$ in this paper. However, I could not find any proof of this formula, and I don't know how to approach to it. Is there good explanation for it?

$$ \pi + 3 = \sum_{n=1}^\infty \frac{n2^nn!^2}{(2n)!} $$

Answer 1

Here's part of an exercise from the book Pi and the AGM by Borwein and Borwein.

Prove that $$\frac{2x\sin^{-1}x}{\sqrt{1-x^2}}=\sum_{m=1}^\infty \frac{m!^2(2x)^{2m}}{m(2m)!}.\tag{1}$$ Hint: show that $f=(\sin^{-1}x)/\sqrt{1-x^2}$ satisfies $(1-x^2)f'=1+xf$.

Granted $(1)$, differentiating and multiplying by $x$ gives a formula for $$\sum_{m=1}^\infty \frac{m!^2(2x)^{2m}}{(2m)!}.$$ Doing it again, gives a formula for $$\sum_{m=1}^\infty m\frac{m!^2(2x)^{2m}}{(2m)!}.$$ Finally set $x=1$.