Is there a closed form for the harmonic series?

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Discussion Overview

The discussion revolves around the question of whether there exists a closed form for the harmonic series, particularly in relation to Euler's approximation involving the natural logarithm and the Euler-Mascheroni constant. Participants explore the definitions and implications of closed forms, asymptotic equivalences, and the role of Bernoulli numbers in this context.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants reference Euler's approximation, suggesting that the harmonic series can be expressed as Σ1/n = ln(n) + γ, questioning if this qualifies as a closed form.
  • Others argue that this expression represents asymptotic equivalence rather than a closed form, emphasizing the need for clarity on what constitutes a closed form.
  • A participant proposes including a summation of Bernoulli numbers divided by powers of n as K approaches infinity, questioning whether this would still be considered a closed form.
  • Concerns are raised about the accuracy of Euler's statement, with one participant asserting that the left-hand side of the equation does not depend on n, thus challenging the validity of the expression.
  • Another participant suggests that while H(n) ~ ln(n) + γ is a better representation, it still does not constitute a closed form, as integrals or sums cannot be classified as such.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether a closed form for the harmonic series exists, with multiple competing views and ongoing debate about the definitions and implications of closed forms and asymptotic relationships.

Contextual Notes

Limitations in the discussion include the ambiguity surrounding the definitions of closed forms and asymptotic equivalences, as well as the unstated domain for the harmonic series summation.

avocadogirl
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Taking from Euler's offering that:

\Sigma1/n= ln(n) + \gamma

could you say that there is a closed form of the harmonic series?

Does Euler's offering qualify?
 
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Your = is not the usual =, but the asymptomic =. As such it is not a closed form in the usual sense. To talk of a closed form one needs to define which functions and operations are to be acceptable and as in this case in what sense two forms are to be equal. The harmonic series in usually not considered to have a closes form as it can not be written in terms of the usual function with the usual =.
 
Could you please explain a little more about this asymptotic equivalence?

Would it be more accurate to include the summation of a "Kth" Bernoulli number, divided by the product of "K" and (n raised to the power of "K"), as K goes from one to infinity?

But, including this summation would not constitute closed form, correct?
 
avocadogirl said:
Taking from Euler's offering that:

\Sigma1/n= ln(n) + \gamma]
No, Euler never said any such thing. The equation you write is obviously false. On the left, you have a sum with n varying over some unstated domain so the sum itself does NOT depend on n, but the right hand side does.

could you say that there is a closed form of the harmonic series?

Does Euler's offering qualify?
 
avocadogirl said:
Could you please explain a little more about this asymptotic equivalence?

Would it be more accurate to include the summation of a "Kth" Bernoulli number, divided by the product of "K" and (n raised to the power of "K"), as K goes from one to infinity?

But, including this summation would not constitute closed form, correct?

H(n)= ln(n) + Euler–Mascheroni
where Hn is the nth harmonic number
would better be written
H(n)~ln(n) + Euler–Mascheroni
to avoid confusion
interpeted as
limit [-H(n)+ ln(n) + Euler–Mascheroni]=0

That is to say that while the two sides are never equal the get closer the large n gets

and no a integral or a sum (finite or infinite) would not be considered a closed form for harmonic numbers
 
Thank you! That clarifies.
 

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