13
votes
2answers
241 views

Closed form of $\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$

What is the closed form of the following integral $$\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx,n\in\mathbb{N}$$ By Mathematica I saw that $$\int_0^\frac{1}{2}x\cot(\pi x)\,dx=\frac{\log(2)}{2\pi}$$ ...
6
votes
3answers
205 views

Question about Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$

For a freshman calculus project, I used Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, and noted from Wikipedia's explanation that the infinite product representation of ...
55
votes
3answers
1k views

A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$

A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help. Prove: ...
16
votes
3answers
303 views

Find the value of $\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx$

I'm trying to figure out how to evaluate the following: $$ J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx $$ I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies ...
4
votes
1answer
150 views

question about Riemann zeta $\zeta (0)$ [duplicate]

i know that $$\zeta (m)=\sum_{n=1}^\infty n^{-m}$$ so $$\zeta (0)=\sum_{n=1}^\infty n^0=1+1+1+1+1+1+\cdots=\infty $$ but actually $$\zeta (0)=-0.5$$ where is the wrong please help thanks ...
1
vote
2answers
112 views

Proof that the zeta function converges for Re(s)>1

It would be absolutely fantastic if anybody could give me some guidance on the question above. For me (please correct me if I'm wrong), this question boils down to proving that ...
7
votes
2answers
322 views

A tough series: $\sum_{k=1}^{\infty}\frac{\zeta(2k)}{(k+1)(2k+1)}$, need help

I was doing a integral which ends up with a tough series part: $$\sum_{k=1}^{\infty}\frac{\zeta(2k)}{(k+1)(2k+1)}$$ Mathematica says $$\frac12$$ Which agrees with the anwer...Anyone know how to ...
3
votes
2answers
144 views

Another improper integral

Show that : $$\int_0^1\frac{(\sin ^{-1}x)^2}{x}\text{d}x=\frac{\pi ^2\ln 2}{4}-\frac78\zeta(3)$$ This integral is in "irresistible integrals" on page 122. I can't prove this one.
2
votes
0answers
190 views

Proof of Euler's general formula for a sum involving harmonic numbers [duplicate]

I have seen this formula, but how to prove this? $$2\sum\limits_{k=1}^\infty \frac{H_k}{\left( k+1 \right)^m} =m\zeta \left( m+1 \right)-\sum\limits_{k=1}^{m-2}{\zeta \left( m-k \right)\zeta \left( ...
9
votes
1answer
244 views

Prove the Wallis formula form $\left(4^{\zeta{(0)}} \cdot e^{-\zeta'{(0)}}\right)^2=\frac{\pi}{2}$

How would you prove the following Wallis formula form $$ \left(4^{\zeta{(0)}} \cdot e^{-\zeta'{(0)}}\right)^2=\frac{\pi}{2}?$$ Thanks in advance!
8
votes
2answers
327 views

How to prove $\zeta'(0)/\zeta(0)=\log(2\pi)$?

How do I prove that $\zeta'(0)/\zeta(0)=\log(2\pi)$ ? I can get $\zeta(0)=-\frac{1}{2}$, but I don't know how to calculate $\zeta'(0)=-\frac{1}{2}\log(2\pi)$ ? Can you help me ? Here $\zeta(s)$ is ...
3
votes
2answers
302 views

Conditional convergence of Riemann's $\zeta$'s series

Do Riemann's zeta-function's partial sums $\sum_{n=1}^N n^{-s}$ converge conditionally for some value $s=\sigma+it$ with $\sigma\le 1$? (We must at least have $t\ne 0$ of course.) Partial summation ...
3
votes
2answers
192 views

Elementary derivation of certian identites related to the Riemannian Zeta function and the Euler-Mascheroni Constant

Is the proof of these identities possible, only using elementary differential and integral calculus? If it is, can anyone direct me to the proofs? ( or give a hint for the solution ) ...
18
votes
3answers
1k views

Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.

Let $ f$ be a function such that $ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $ (f is $2\pi$-periodic) such that $ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$ Computing the Fourier series of $f$ and ...
12
votes
1answer
400 views

challenging integral involving $\zeta(5)$

I ran across a curious integral that seems to be rather tough that some on the site may enjoy. Show that $$\displaystyle \int_{0}^{1}\frac{\sqrt{1-x^{2}}}{1-x^{2}\sin^{2}(x)}dx = ...
10
votes
2answers
472 views

Integrating $\frac{x^k }{1+\cosh(x)}$

In the course of solving a certain problem, I've had to evaluate integrals of the form: $$\int_0^\infty \frac{x^k}{1+\cosh(x)} \mathrm{d}x $$ for several values of k. I've noticed that that, for k a ...
33
votes
4answers
5k views

Nice proofs of $\zeta(4) = \pi^4/90$?

I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute ...