I have been looking into zeta sums and whatnot lately and realised the following equality. $$\frac{\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{3}}}{\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}} = \sum_{n=0}^{\infty}\frac{\frac{(-1)^{n}}{(2n+1)^{3}}}{\frac{(-1)^{n}}{2n+1}}$$ Note the values of the series are: $$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{3}} = \frac{\pi^3}{32},\quad\quad \sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1} = \frac{\pi}{4}, \quad\quad \sum_{n=0}^{\infty}\frac{1}{(2n+1)^{2}} = \frac{\pi^{2}}{8}$$ I assume this is a coincidence but it would be very nice to know if it's not. Are there other examples of series with this property?
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Doesn't it follow from functional equation for Dirichlet L-series? – xsnl Mar 26 '18 at 18:08
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isn't this something in the direction you are intersted in? https://math.stackexchange.com/questions/2454168/proving-that-two-real-numbers-are-equal/2461582#2461582 – tired Mar 26 '18 at 18:28