Negative powers of 4

Algebra Level pending

Let \(A = 1^{-4}+2^{-4}+3^{-4 }+4^{-4 }+5^{-4}+\cdots\) denote the sum of the reciprocals of the fourth powers of all positive integers, and \(B = 1^{-4 }+ 3^{-4 }+ 5^{-4 }+ 7^{-4 }+ \cdots \) a similar sum for all odd positive integers.

Let \( \dfrac AB = \dfrac CD\), where \(C\) and \(D\) are coprime positive integers, find \(C+D\).

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