Double sums that result Catalan's constant.

Hello brilliants, while playing around with the problems here 1 and here 2, I found Catalan's Constant, G by making few change in those problems. ie $\text{Prove that Catalan's Constant }$ $G= \sum_{m=1}^{\infty}\sum_{n=0}^{\infty} \frac{1}{(2m+n)^2}\left((-1)^m-(-1)^{m+n}+(-1)^n\left(1+\frac{2m}{(m+n)}+\frac{m^2}{(m+n)^2}\right)\right)$

I shared this problem to Romanian mathematical Magazine as a proposal and this was proved by Sergio Esteban who came solution with integral solution. I approached to result with and without integral work and I'm looking forward for such more works from brilliants. The solutions here presented will be highly appreciated. :)

Thank you!!

#Calculus

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$