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?
Exact value of an infinite sum
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$\begingroup$
sequences-and-series
convergence
closed-form
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1https://www.wolframalpha.com/input/?i=sum(n%3D0,infinity,1%2F(2%5En%2B1)) – 2017-01-22
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1I don't think it's any easier than finding a closed form of the Erdős-Borwein constant... – 2017-01-22
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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](http://mathworld.wolfram.com/q-PolygammaFunction.html). – 2017-01-23
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0see also http://math.stackexchange.com/questions/662795/closed-form-solution-to-infinite-sum – 2017-01-23
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1Just as Lucian commented $$\sum_{n=0}^\infty \dfrac1{a^n+b}=\frac{\psi _{\frac{1}{a}}^{(0)}\left(-\frac{\log (-b)}{\log (a)}\right)-\log \left(\frac{a}{a-1}\right)}{b \log (a)}$$ – 2017-01-23