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Some deleted answers! Some of them are accepted and the other are correct. I just would like to keep them at hand.

(1)- (2)- (3)- (4)- (5).

Is there a closed form for $\sum_{j=1}^{n} j^2\log{j}$?

$$ -\zeta' \left( -2 \right) +\zeta_1' \left( -2,n+1 \right). $$

A more general case

$$ \sum_{k=1}^{n} j^{s}\ln(j)^{m}= \lim_{\alpha \to s } \zeta^{(m)}_{\alpha}( - \alpha ) + \zeta^{(m)}_{\alpha} \left( -\alpha,n+1 \right). $$


Sep
13
comment Find $\int \sin^{-1}\sqrt{x}\hspace{1mm}dx$
Have you tried integration by parts?
Sep
13
revised Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$
adding alink
Sep
13
revised Does the series $\sum_{i=0}^{\infty}\exp\{{-(m!)!}\}(D^m\phi)(0)$ converge for every $\phi \in C^\infty$?
language
Sep
13
comment How to prove a limit exists using delta and epsilon?
@user175623: You are welcome.
Sep
13
answered How to prove a limit exists using delta and epsilon?
Sep
13
revised Given $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$
language
Sep
13
comment How to show this integral equals $\pi^2$?
@mike: You are welcome.
Sep
13
comment How to show this integral equals $\pi^2$?
@mike: I fixed it. Thanks for the comment.
Sep
13
revised How to show this integral equals $\pi^2$?
typo
Sep
13
comment How to show this integral equals $\pi^2$?
@mike: Yes, because we want to generate the $\ln(x)$ under the integral sign. See the original integral and the links.
Sep
13
comment How to show this integral equals $\pi^2$?
@mike: We just differentiate with respect to $s$. You know the derivative of $x^{s-1}$ with respect to $s$.
Sep
13
comment Compute $\sum\limits_{m=0}^{\infty}\frac{2^{m+1}}{i\lambda}(1+e^{-i\lambda\frac{2}{2^{m+1}}}-2e^{-i\lambda\frac{1}{2^{m+1}}})$
By the way where did this series come from?
Sep
13
revised Why does this function decrease at this speed?
typo
Sep
13
revised $\int_0^{\infty}\frac{\ln x}{x^2+a^2}\mathrm{d}x$ Evaluate Integral
adding alink
Sep
13
revised How to show this integral equals $\pi^2$?
improved formatting
Sep
13
comment How to show this integral equals $\pi^2$?
@VitalStatistix: You are very welcome.
Sep
13
comment How to show this integral equals $\pi^2$?
@Gahawar: Thank you for the comment. I really appreciate it.
Sep
13
answered How to show this integral equals $\pi^2$?
Sep
12
revised Why does this function decrease at this speed?
improved formatting
Sep
12
comment Intuitively, how do you explain the concept of Flux?
Flux is the flow of a physical property in space. For instance the heat flux from a hot surface.