s-n-1-1-n-1-n-s-Dirichlet-eta-function-prove-that-s-1-2-1-s-s-

Question Number 2815 by prakash jain last updated on 28/Nov/15

η (s) = \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n - 1}}{n^{s}} Dirichlet eta function

prove that

η (s) = (1 - 2^{1 - s}) ζ (s)

Commented by prakash jain last updated on 27/Nov/15

This result is used in answer to Q 2796 .

Commented by 123456 last updated on 28/Nov/15

η (s) = \underset{n = 1}{\sum^{+ \infty}} \frac{{(- 1)}^{n - 1}}{n^{s}} (? ? ?)

Commented by prakash jain last updated on 28/Nov/15

Yes . corrected .

Answered by prakash jain last updated on 28/Nov/15

For s \in R, s > 1

η (s) = \frac{1}{1^{s}} - \frac{1}{2^{s}} + \frac{1}{3^{s}} - \frac{1}{4^{s}} + . .

η (s) = \frac{1}{1^{s}} + (\frac{1}{2^{s}} - \frac{2}{2^{s}}) + \frac{1}{3^{s}} + (\frac{1}{4^{s}} - \frac{2}{4^{s}}) + \dots

t a k i n g a l l - v e t e r m s t o w a r d s e n d .

s e r i e s r e a r r a n g e m e n t i s v a l i d f o r s > 1 .

η (s) = ζ (s) - \frac{2}{2^{s}} [\frac{1}{1^{s}} + \frac{1}{2^{s}} + \dots]

η (s) = ζ (s) - 2^{1 - s} ζ (s)

η (s) = ζ (s) (1 - 2^{1 - s})