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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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  • $\begingroup$ Doesn't it follow from functional equation for Dirichlet L-series? $\endgroup$
    – xsnl
    Commented Mar 26, 2018 at 18:08
  • $\begingroup$ isn't this something in the direction you are intersted in? math.stackexchange.com/questions/2454168/… $\endgroup$
    – tired
    Commented Mar 26, 2018 at 18:28

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