Linked Questions

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...

Is there an exact form of $$f(s)=\sum_{n=0}^\infty {\frac{(-1)^n}{(2n+1)^s}}=1-\frac{1}{3^s}+\frac{1}{5^s}-\frac{1}{7^s}+\dots$$ when $s$ is odd? Discussion I have been exploring infinite series ...

Prove that : $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^3}=\frac{\pi^3}{32}.$$ I think this is known (see here), I appreciate any hint or link for the solution (or the full solution).

After numerical analysis it seems that $$ \frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}} $$ Could someone prove the validity of such identity?

Hi I am trying to prove the relation $$ I:=\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}. $$ I tried expanding the log argument by using $\sin x/ \cos x=\tan x,$ and than used $\log(...

I was trying to compute the integral: $$ I = \int_0^1 \frac{(\operatorname{ln}x)^4}{1+x^2} dx$$ This is not a very difficult integral to evaluate if one knows the standard procedure used in evaluating ...

I've been trying to find a way to compute the integral $$\int_{0}^{\infty}\frac{1}{x^a + 1} dx, \quad a > 1$$ without having to resort to the heavy machinery of complex analysis. I've found two ...

I am trying to solve the following: The Euler numbers $E_n$ are defined by the power series expansion $$\frac{1}{\cos z}=\sum_{n=0}^\infty \frac{E_n}{n!}z^n\text{ for }|z|<\pi/2$$ (a) Show that $...

Let $P(z)$ and $Q(z)$ be polynomials such that the degree of $Q(z)$ is exactly one degree more than the degree of $P(z)$. And assume that $ \displaystyle \sum_{n=-\infty}^{\infty} (-1)^{n} \frac{P(n)}{...

How would I calculate $$\mathrm{Res}\left(\frac{\pi}{\sin(\pi z)(2z+1)^3}\right)?$$ I understand it has singularities at $z=n$ and $z=-1/2$, I'm interested in the residue when $z=-1/2$. I know that ...

Latest Edit Thanks to Mr Ali Shadhar who gave a beautiful closed form of the derivative as $$\boxed{S_n = \frac { \pi ^ { 2 n + 1 } } { 4 ^ { n + 1 } ( 2 n ) ! } | E _ { 2 n } | }, $$ where $E_{2n}$ ...

$$S=\sum_{k=1}^{\infty}\left(\frac{\left(-1\right)^{k-1}}{k+\frac{1}{2}}\ln\left(1-\frac{1}{\left(k+\frac{1}{2}\right)^{2}}\right)\right)$$ Let, $$f(a)=\sum_{k=1}^{\infty}\left(\frac{\left(-1\right)^{...

I am trying to find the $n$-th derivative of $\csc(m\pi)$, so I took few cases: for simplicity let $x=\cot(m\pi)$ and $y=\csc(m\pi)$, $$\frac{d^0}{dm^0}\csc(m\pi)=\pi^0(\color{red}{1}x^0y^1)$$ $$\frac{...

I would like to find a proof for the generating formula for odd values of Dirichlet beta function, namely: $$\beta(2k+1)=\frac{(-1)^kE_{2k}\pi^{2k+1}}{4^{k+1}(2k)!}$$ My try was to start with the ...

This is an identity from Ramanujan's letter, I am just curious. How do you prove this. My math level knowledge is still very basic so a simplified proof is preferred: $$\frac{1}{1^{5}\cosh(\frac{\pi}{...

How can I expand $\dfrac{\pi \csc(z\pi)}{(2z+1)^3}$? so then I can find the residue ? thanks

I need hints on the proof of: $$\int_0^\infty\dfrac{\ln(x)^2}{1+x^2}{\rm{d}x}=\dfrac{\pi^3}{8}$$ and then: $$\sum\limits_{n=0}^\infty\left((-1)^n\dfrac{1}{(2n+1)^3}\right)=\dfrac{\pi^3}{32}$$ ...