Find the exact value of $$\sum_{n=0}^\infty \frac{1}{2^n+1}$$ in terms of well known functions, constants, etc.
Since every term is less than the corresponding term in the powers of $2$ series, the series converges, but what is its exact value?
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1https://www.wolframalpha.com/input/?i=sum(n%3D0,infinity,1%2F(2%5En%2B1)) – Peter Jan 22 '17 at 21:00
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1I don't think it's any easier than finding a closed form of the Erdős-Borwein constant... – Gabriel Romon Jan 22 '17 at 21:08
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1In general, series of the form $\displaystyle\sum_{n\geqslant0}\dfrac1{a^n+b}$ possess a closed form in terms of the q-polygamma function. – Lucian Jan 23 '17 at 00:00
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see also http://math.stackexchange.com/questions/662795/closed-form-solution-to-infinite-sum – 926reals Jan 23 '17 at 02:52
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1Just as Lucian commented $$\sum_{n=0}^\infty \dfrac1{a^n+b}=\frac{\psi _{\frac{1}{a}}^{(0)}\left(-\frac{\log (-b)}{\log (a)}\right)-\log \left(\frac{a}{a-1}\right)}{b \log (a)}$$ – Claude Leibovici Jan 23 '17 at 07:51