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

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 $x=1$.
