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the mean square The Mean Square
(with one standard deviation and several unusual ones)

aka Rob Johnson

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2d
answered Integral of ln(x)sech(x)
2d
answered Formulating a recursive definition
2d
comment Finding the Moment Generating function of a Binomial Distribution
You are missing an $e^{tx}$ in the first line.
2d
answered Finding the Moment Generating function of a Binomial Distribution
2d
comment Evaluate$ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) dx$
The simplest method seems to be to use an identity that I've used quite a bit: $$\log(1+\cos(x))=2\sum_{k=1}^\infty(-1)^{k-1}\frac{\cos(kx)}{k}-\log(2)$$ Integrating this and knowing Catalan's Constant finishes things up. I use these in my answer.
2d
awarded  Necromancer
2d
comment $\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2)$
I forgot that you had used the generating function of the Harmonic Numbers here. I used its integral for the sum of $H_n^2/n^2$ recently.
2d
answered Evaluate$ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) dx$
2d
comment How to solve $e^x=x$?
@JackD'Aurizio: do you have any estimates from the integral of how many roots there are inside a given region (circle, square, whatever)?
2d
comment How to solve $e^x=x$?
@XiangruLian: This formula works if the roots are all simple. Multiple roots are counted with their multiplicity.
2d
answered How to solve $e^x=x$?
Aug
26
comment If $\left(1^a+2^a+\cdots+n^{a}\right)^b=1^c+2^c+\cdots+n^c$ for some $n$, then $(a,b,c)=(1,2,3)$?
For some $n\gt1$, I assume.
Aug
26
awarded  summation
Aug
25
comment How is $ i^{-1} = -i$ and $i^{-3} = i$?
This has drifted far from the topic. Soon it will be time to remove a good portion of these comments.
Aug
25
comment How is $ i^{-1} = -i$ and $i^{-3} = i$?
$\sqrt{-1}$ is not well-defined. The square root function can be extended to the complex plane with an appropriate branch cut, but depending on the branch cut used, $\sqrt{-1}$ could be $i$, $-i$, or simply not defined (when the branch cut contains $-1$). Because of the lack of a conventional definition, it is better to use $i$ or $-i$, depending on which is intended.
Aug
25
answered 'Proof ' that $\ln(x)$ converges
Aug
25
answered $(a_n)$ is bounded $\implies \sum a_n \frac{1}{n^2}$ converges
Aug
25
comment How to deduce a closed formula given an equivalent recursive one?
actually, both of the last two identities are the generalized binomial theorem.
Aug
25
revised How to deduce a closed formula given an equivalent recursive one?
add some more explanation
Aug
25
comment How to show that the Laurent series of the Riemann Zeta function has $\gamma$ as its constant term?
@Henry: When $s\in\mathbb{R}$, $n^{-s}$ follows a straight line (along the real axis) from $1$ to $0$, as opposed to the case where $s\not\in\mathbb{R}$, in which $n^{-s}$ spirals from $1$ to $0$ (which is discussed in the next sentence). Remember we are assuming $\mathrm{Re}(s)\gt0$.