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 Jun 17 comment Prove the convergence/divergence of $\sum \limits_{k=1}^{\infty} \frac{\tan(k)}{k}$ @Chris aren't you confusing limit of 'function' with limit of a sequence? Jun 17 comment The series $\sum\limits_{k=1}^{\infty} \frac1{\sqrt{{k}{(k^2+1)}}}$ @Chris sure, that's very likely. But even though having a good approximation in nice closed form is worth a shot :) Jun 17 comment The series $\sum\limits_{k=1}^{\infty} \frac1{\sqrt{{k}{(k^2+1)}}}$ @Chris you can also use integral to estimate the sum: $$\int_{\frac{1}{2}}^{+\infty} \frac{dx}{\sqrt{x(x^2+1)}}$$ Mathematica gives the answer in terms of hypergeometric function: $2 \sqrt{2} \, _2 F_1 \left( \frac{1}{4}, \frac{1}{2} ; \frac{5}{4}, -4 \right) \approx 2.3261$ while the sum is around $2.2641$. Jun 17 comment The function $f(x) = \int_0^\infty \frac{x^t}{\Gamma(t+1)} \, dt$ I think it is Ramanujan's formula, i.e.: $$\int\limits_{0}^{\infty}\frac{z^{t} \, dt}{\Gamma(1+t)}=e^{z}-\int\limits_{0}^{\infty}\frac{e^{-z\tau}d\tau}{\tau(\ln^2 (\tau)+\pi^{2})},\,\,\, Re(z)\ge{0}$$ Jun 15 comment Evaluting: $\int\frac{1}{(1+\tan x)^2} dx$ @experimentX $$\frac{\sin x}{ \sin x + \cos x} - \frac{-\cos x}{\sin x + \cos x} = 1 \neq 2$$ ;) Jun 15 answered Evaluating $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\exp(n x-\frac{x^2}{2}) \sin(2 \pi x)\ dx$ Jun 15 awarded Enthusiast Jun 13 comment Inequality between volume and its projections @nikita2 even obvious facts need a proof in mathematics :) Jun 13 comment Inequality between volume and its projections @Potato sure it should be measurable. Jun 13 awarded Scholar Jun 13 accepted Inequality between volume and its projections Jun 13 awarded Student Jun 13 asked Inequality between volume and its projections Jun 12 comment Do improper integrals like $\int_{-\infty}^{+\infty} f$ converge if $xf(x)\rightarrow 0$? In general the function you're integrating can have no limit at infinity. The same holds for $x f$. Take for example comb function d.pr/i/ElNd+ or if you want to deal with formulas, take sth like $f(x) = x^2 \exp (-x^8 \sin^2 (40x))$. Jun 12 awarded Critic Jun 12 comment Limit finding of an indeterminate form @Gigili just a typo :) Jun 12 revised Limit finding of an indeterminate form edited body Jun 12 comment Limit finding of an indeterminate form @AlexChamberlain take for example: $\lim_{x\to 0} \frac{\sin x}{x}$. When you're applying l'Hopital's rule, you're using the fact that $(\sin x)' = \cos x$ but it's the consequence of $\frac{\sin x}{x} \to 1$ when $x\to 0$. Jun 12 answered Limit finding of an indeterminate form Jun 10 awarded Commentator