Does a closed form formula exist for the following series?
$\sum\limits_{n = 1}^N {\frac{{1 + {x^{2n - 1}}}}{{1 + {x^{2n + 1}}}}}$
2026-02-22 21:21:50.1771795310
Does a closed form formula for the series $\sum\limits_{n = 1}^N {\frac{{1 + {x^{2n - 1}}}}{{1 + {x^{2n + 1}}}}}$ exist?
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There is "closed" form expression $$\sum\limits_{n = 1}^p {\frac{{1 + {x^{2n - 1}}}}{{1 + {x^{2n + 1}}}}}=p+\frac{\left(1-x^2\right) }{x^2 \log \left(x^2\right)}\left(\psi _{x^2}^{(0)}\left(p+1-\frac{\log \left(-\frac{1}{x}\right)}{\log \left(x^2\right)}\right)-\psi _{x^2}^{(0)}\left(1-\frac{\log \left(-\frac{1}{x}\right)}{\log \left(x^2\right)}\right)\right)$$ where appears the $q$ digamma function (have a look here).