Sum of Reciprocals of Cubes of Odd Integers Alternating in Sign in Pairs

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Theorem

\(\displaystyle \frac 1 {1^3} + \frac 1 {3^3} - \frac 1 {5^3} - \frac 1 {7^3} + \cdots\) \(=\) \(\displaystyle \frac {3 \pi^3 \sqrt 2} {128}\)


Proof


Sources