Resummation of the Harmonic series

In summary, there are different ways to sum Cauchy-divergent series to finite values, but it is unclear if such a procedure exists for the Harmonic series. There is a connection with the Riemann zeta-function's pole at unity, but this does not provide a basic zeta regularization for the Harmonic series. The only known way to assign a value to the Harmonic series is through the Ramanujan sum of Euler's constant, which is not very impressive since it is just the constant of the series in the Euler-Maclaurin sum formula.
  • #1
yasiru89
107
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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)
 
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  • #2
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)
 
  • #3
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.
 

1. What is the Harmonic series?

The Harmonic series is a mathematical series that is defined as the sum of the reciprocals of positive integers. It is written as 1 + 1/2 + 1/3 + 1/4 + ... and is known to diverge, meaning that the sum reaches infinity as more terms are added.

2. Why is the Harmonic series important?

The Harmonic series is important in mathematics because it has connections to many other mathematical concepts, such as prime numbers and the distribution of prime gaps. It also has applications in physics, particularly in quantum field theory and the study of black holes.

3. What is "resummation" of the Harmonic series?

Resummation of the Harmonic series is a mathematical technique used to find a finite value for the sum of the series, even though it diverges. This is done by regrouping the terms in a different way and using mathematical tools such as the Euler-Maclaurin formula or zeta function regularization.

4. Why is resummation of the Harmonic series useful?

Resummation of the Harmonic series allows us to assign a finite value to a series that would otherwise diverge. This can be useful in situations where the series appears in a physical theory or calculation, and a finite result is needed. It also helps us gain a better understanding of the underlying mathematical structures and connections between seemingly unrelated series.

5. Are there any practical applications of resummation of the Harmonic series?

Yes, there are practical applications of resummation of the Harmonic series. One example is in the calculation of the critical temperature in statistical mechanics, where the resummation of the series gives a more accurate result compared to a naive summation. It is also used in other areas of physics, such as in the study of phase transitions and the renormalization group.

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