Inspired by Calvin Lin

Calculus Level 3

\[\begin{eqnarray} &&\left(\frac{1}{4^2}+\frac{1}{8^2}+\frac{1}{12^2}+\ldots\right) \\& + &\left(\frac{1}{4^4}+\frac{1}{8^4}+\frac{1}{12^4}+\ldots\right) \\ & + & \left(\frac{1}{4^6}+\frac{1}{8^6}+\frac{1}{12^6}+\ldots\right) \\ & + & \ldots \end{eqnarray} \]

What is the sum of this series?

Note that the bases are divisible by 4, and the exponents are even.

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