Re: harmonic series convergence

From: Zdislav V. Kovarik (kovarik_at_mcmaster.ca)
Date: 08/23/04 

Date: Mon, 23 Aug 2004 18:41:47 -0400

On Mon, 23 Aug 2004, Narasimham G.L. wrote:

> To what limit does H(n)= Sum{k=1..Inf}[(-1)^(k-1)/k^n] converge? (In
> Riemann zeta function signs of alternate terms have been altered). TIA

(Assuming n>1, or for complex n, Real(n) > 1)

Request: Don't denote it H(n); this is customary for the true harmonic
numbers

 H(n) = 1 + 1/2 + ... + 1/n .

Let me call the alternating sibling of Zeta function A(n),
hoping I am not trespassing on anyone's favorite use of it.

An exercise for you: Add up the terms carrying minus signs, factor out
a suitable power of 2, and relate the result to zeta function. Then
restore the alternating series.

Remark: If you express Zeta(n) in terms of A(n), you will obtain an
analytic continuation of Zeta to Real(n) > 0, except of course the
case n=1.

Cheers, ZVK(Slavek).