Show that $(1-2^{1-z})\zeta(z)=\sum\limits_{s=1}^\infty \frac{(-1)^s}{s^z}=$ for Re$(z)>0$.

Asked Apr 18 '16 at 00:44

Active Apr 18 '16 at 00:44

Viewed 88 times

Show that $(1-2^{1-z})\zeta(z)=\sum\limits_{s=1}^\infty \frac{(-1)^s}{s^z}=\frac{1}{\Gamma(z)}\int\limits_{0}^\infty \frac{t^{z-1}}{e^t+1}dt$ for Re$(z)>0$.

Not sure how to get started on this, we have covered some of the formulas for the $\zeta$ function but I don't really have any intuition regarding them.

asked Apr 18 '16 at 00:44

Jürgen Sukumaran

7,645

1

see the hint I wrote there : http://math.stackexchange.com/questions/1746711/alternating-series-test-for-complex-series/1746981#1746981 (Last way) – reuns Apr 18 '16 at 00:47
1

I showed that this is true using the Generalized Dirichlet Test in THIS ANSWER – Mark Viola Apr 18 '16 at 03:41

Show that $(1-2^{1-z})\zeta(z)=\sum\limits_{s=1}^\infty \frac{(-1)^s}{s^z}=$ for Re$(z)>0$.

0 Answers0