Hard product!

$\large \displaystyle\prod _{ n=1 }^{ \infty }{ \dfrac { 1 }{ e } \left( \dfrac { 2n }{ 2n-1 } \right) ^{ (4n-1)/2 } } =\dfrac { { A }^{ 3 } }{ { \pi }^{ 1/4 }{ 2 }^{ 7/12 } }$

Prove that the equation above holds true, where $A$ denotes the Glaisher–Kinkelin constant.

This is a part of the set Formidable Series and Integrals

#Calculus

Easy Math Editor

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`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`	$\sum_{i=1}^3$
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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}`	$\boxed{123}$