A neat enigma

Calculus Level 3

\( \displaystyle \sum _{ n=1 }^{ \infty }{ \frac { 1 }{ a{ n }^{ 2 } } =\int _{ 0 }^{ 1 }{ \frac { dx }{ \sqrt { 4-{ x }^{ 2 } } } } } \)

Find the value of \(a\).


This problem is original.

Picture credits: Rainbow 02 by Jerry Magnum Porsbjer, Wikipedia

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