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\(\sum_{n=2}^{\infty} \frac{(-1)^n}{n\log{n}}\)

Posted: Sat Mar 04, 2017 1:12 pm
by Grigorios Kostakos
Evaluate the convergent series \[\mathop{\sum}\limits_{n=2}^{\infty} \frac{(-1)^n}{n\log{n}}\,.\]


Note: I don't have a solution for this.

Re: \(\sum_{n=2}^{\infty} \frac{(-1)^n}{n\log{n}}\)

Posted: Sat Dec 14, 2019 3:16 pm
by Riemann
Basically it equals to

$$\int_1^\infty \left( 1+\left(2^{1-s}-1\right)\zeta(s) \right) \, \mathrm{d}s$$

However, the $\zeta$ function does not behave well under integrals. So, I would not expect a closed form to exist ... !