Interesting Factorial Sum

I wanted to share this sum with you all, though I have no clue how to prove it ---

$\sum_{n=1}^{\infty}\left[(-1)^n \dfrac{(n!)^2}{n^2(2n)!}\right]=2\log(\phi)\log\left(\dfrac{1}{\phi}\right)$ where $\phi$ is the golden ratio.

It has an amazing closed form, doesn't it?

#Calculus #Summation #Factorials

Easy Math Editor

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Comments

http://mathworld.wolfram.com/CentralBinomialCoefficient.HTML gives a bit of info

Bogdan Simeonov - 6 years, 7 months ago

Here's a proof for a lot of these( including this one ): click me

Bogdan Simeonov - 6 years, 7 months ago

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}`	$\boxed{123}$