Lucian
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25,986
90/100 score
 1h comment How to find $\int \frac{\ln(x)}{x^2}dx$ As an aside, for $n>-1$ we have $~\displaystyle\int_1^\infty\frac{\ln^n(x)}{x^2}~dx~=~n!$ 3h awarded Nice Answer 14h comment a function defined as an integral can be continued analytically In general, $~\displaystyle\int_0^\infty\frac{t^n}{e^t+1}~dt~=~n!~\eta(n+1),~$ and $~\displaystyle\int_0^\infty\frac{t^n}{e^t-1}~dt~=~n!~\zeta(n+1),~$ see the Dirichlet $\eta$ and Riemann $\zeta$ functions for more information, both of which possess well-known analytic continuations. 14h revised compute $\lim_{n\rightarrow\infty}\sum_{k=1}^n \sin(\pi \sqrt{k/n}) (1/\sqrt{kn})$ added 1 character in body 14h answered compute $\lim_{n\rightarrow\infty}\sum_{k=1}^n \sin(\pi \sqrt{k/n}) (1/\sqrt{kn})$ 14h comment compute $\lim_{n\rightarrow\infty}\sum_{k=1}^n \sin(\pi \sqrt{k/n}) (1/\sqrt{kn})$ I believe you forgot a factor. 16h comment Find the sub-area of a circle cut by chords Hint: Divide the region into a triangle and a slice. 20h comment Prove the Maclaurin-series representation of $\sqrt{1 + x}$. See binomial series. 20h comment $\sum\limits _{n=1}^{\infty}\frac{1}{n^s}=50$ Riemann-Zeta @robjohn: other that perhaps pulling out a CAS - Precisely. There is no closed form for it. Numerical algorithms are the only way out. 21h answered $\sum\limits _{n=1}^{\infty}\frac{1}{n^s}=50$ Riemann-Zeta 21h comment Integral involving Bessel functions of the first kind @Chappers: That book is well above my pay-grade. :-$)$ I was hoping for an easier approach, one that even the likes of me could understand. :-$)$ 23h comment Is $f(z)=\sum_{k=1}^{\infty}\frac{1}{(k+\frac{1}{k})^{z}}$ somehow related to Riemann's zeta function? $g(x)=x$ and $h(x)=x+\dfrac1x$ are two wholly different geometric shapes. The former is a straight line, the latter a hyperbola. 23h comment Is $f(z)=\sum_{k=1}^{\infty}\frac{1}{(k+\frac{1}{k})^{z}}$ somehow related to Riemann's zeta function? No, there is no meaningful connection between the two, other than their superficial similarity in both definition and numerical value. If anything, I would search for a connection to the polygamma function (a generalization of harmonic numbers), as well as trigonometric and hyperbolic functions. 23h comment Integral involving Bessel functions of the first kind I posted a question asking for a proof of the above identity weeks ago, but found no satisfactory answer. 1d comment Integral involving Bessel functions of the first kind $I~=~\dfrac2{3\pi}.$ 1d answered Lambert W-Function 1d comment Exploring $\sum_{n=0}^\infty \frac{n^p}{n!} = B_pe$, particularly $p = 2$. Hint: Since $n!~=~1\cdot2\cdot3\cdots(n-1)~n$, an obvious approach would be rewriting $n^2$ as $n~(n-1)+n$. 1d comment solving the integral of $e^{x^2}$ See error function, Liouville's theorem, and the Risch algorithm. 1d comment holomorphic function writen as a serie $f_a(z)=(1+z)^a.~$ See binomial series. 2d answered Evaluate $-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$ in $\gamma$.