Register to reply

Resummation of the Harmonic series!

by yasiru89
Tags: harmonic, resummation, series
Share this thread:
Oct26-08, 02:12 AM
P: 107
We have different summation senses under which Cauchy-divergent series can be summed to finite values. I was wondering if such a procedure existed for the Harmonic series, [itex]\sum_{n = 1}^{\infty} n^{-1}[/tex].

I'm putting this in the number theory discussion since the obvious connection with the Riemann zeta-function's pole at unity. However this guarantees there's no basic zeta regularization to the harmonic- so is there a deeper zeta-based result? Or some other way in which a value might be assigned to it? (the only one I know of is the Ramanujan sum of [itex]\gamma[/tex], which is just based on the definition of Euler's constant in the case of the harmonic- and expressing this in terms of other constants has so far proven futile)
Phys.Org News Partner Science news on
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
Oct26-08, 03:01 AM
P: 193
of course Yasiru since [tex] \zeta (1) [/tex] is infinite the regularization procedure is useless, this is a pain in the neck but can be solved via ramanujan summation

[tex] S = \sum_{n=1}^{N}a(n)- \int_{1}^{N} dx a(x) [/tex]

and taking N--->oo if you set a(n)=1/n (Harmonic series) you would get

[tex] \sum_{n=1}^{N}1/n = \gamma [/tex] (Euler's constant)
Oct26-08, 06:58 AM
P: 107
I wonder though- is that the only one that works for the harmonic? Its not as impressive as it could be since we end up with the definition of Euler's constant as the constant of the series in the Euler-Maclaurin sum formula.

Register to reply

Related Discussions
Divergent Harmonic Series, Convergent P-Series (Cauchy sequences) Calculus & Beyond Homework 1
Harmonic or p-series Calculus & Beyond Homework 3
I have a problem with the Harmonic Series, please School me Calculus & Beyond Homework 6
Proof of Divergence for the Harmonic Series. Calculus & Beyond Homework 2
Divergence of the harmonic series Fun, Photos & Games 0