The series $$\sum_{k=1}^\infty \frac{1}{k^k}$$ clearly converges. Is it possible to compute its value analytically?
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1This is Sophomore's Dream. Have a look at https://en.wikipedia.org/wiki/Sophomore's_dream – LuxGiammi Mar 13 '18 at 19:06
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Another one: https://math.stackexchange.com/questions/21330/closed-form-for-sum-frac1nn: – Martin R Mar 13 '18 at 19:07
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Since such series equals $\int_{0}^{1}\exp\left(-x\log x\right),dx$, its numerical value turns out to be pretty close to $$ \exp\int_{0}^{1}-x\log(x),dx = e^{1/4},$$ by Jensen's inequality. – Jack D'Aurizio Mar 13 '18 at 19:13
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That is an interesting estimate, thanks! – MSDG Mar 13 '18 at 19:29