# Tagged Questions

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### Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\,\epsilon>0$ converge?

Numerical results showed that the series $$Q=\sum_{n=1}^{\infty} (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\epsilon>0\tag{1}$$ with $\epsilon=10^{-6}$ converged to $-0.53259554096828...$ We can ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
### 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 ...