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 17h comment How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$? Or more generally $$1\leq\ n^{\frac{1}{n}}\le (1+\epsilon)^{n^{\large \eta-1}}, \ \epsilon>0, 1>\eta>0$$ 17h comment How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$? @Did Right. Alternatively, having in mind a similar idea, we see that $$1\leq\ n^{\frac{1}{n}}\le (1+\epsilon)^{\sqrt{n}/n}, \ \epsilon>0$$ for some $n\ge n_0$. Taking the limit as $n \to \infty$ we're done. 1d comment Another beautiful integral (Part 2) A loooonnnnnnnnnnnggggggggggggg answer. Good job though!:-) (+1) Oct 1 comment Another beautiful integral (Part 2) @nospoon you made a lot of progress. Good job!(+1) Oct 1 comment Another beautiful integral (Part 2) @nospoon Try to add some spaces in the text and it will be fixed. Sep 30 comment Another beautiful integral (Part 2) @tired maybe there are very nice and short ways to do it. I feel I miss a nice way to go. Still working on it here. Sep 30 comment Another beautiful integral (Part 2) @nospoon very nice paper. Thanks (+1) Sep 30 comment Another beautiful integral (Part 2) @coffeemath also with Mathematica things become pretty complicated after $2$ integrations.The cases $n=2, 3$ are easy, but when dealing with $n\ge4$ the job to do is pretty difficult. Sep 30 revised Another beautiful integral (Part 2) added 190 characters in body Sep 30 asked Another beautiful integral (Part 2) Sep 27 accepted Another beautiful arctan integral $\int_{1/2}^1 \frac{\arctan\left(\frac{1-x^2}{7 x^2+10x+7}\right)}{1-x^2} \, dx$ Sep 27 comment Another beautiful arctan integral $\int_{1/2}^1 \frac{\arctan\left(\frac{1-x^2}{7 x^2+10x+7}\right)}{1-x^2} \, dx$ After cancelation, it remains $$\frac{1}{2}\left(\operatorname{Ti}_2\left(\frac{1}{6}\right)-\operatorname{Ti}‌​_2\left(\frac{1}{9}\right)\right)$$ I think. Sep 27 comment Another beautiful arctan integral $\int_{1/2}^1 \frac{\arctan\left(\frac{1-x^2}{7 x^2+10x+7}\right)}{1-x^2} \, dx$ (+1) for telling what Maple says. Sep 27 revised Another beautiful arctan integral $\int_{1/2}^1 \frac{\arctan\left(\frac{1-x^2}{7 x^2+10x+7}\right)}{1-x^2} \, dx$ added 1275 characters in body Sep 27 asked Another beautiful arctan integral $\int_{1/2}^1 \frac{\arctan\left(\frac{1-x^2}{7 x^2+10x+7}\right)}{1-x^2} \, dx$ Sep 23 awarded Nice Question Sep 21 answered Does $\sum_{n\geq 2} \dfrac{\ln(1+n)}{\ln(n)}-1$ converge/diverge? Sep 16 awarded Nice Question Sep 16 comment Simplification of an expression involving the dilogarithm with complex argument @DavidH (+1) good job! I'll also think of this way and see if it can be simplified further. Sep 15 awarded Popular Question