What is the Function for the Value of a Convergent Series Sum?

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    Convergent Series Sum
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Discussion Overview

The discussion revolves around the convergence of the series ##\sum_n \frac{1}{n^c}## for values of ##c > 1## and whether there exists a function that expresses the value of this sum as a function of ##c##. The scope includes theoretical aspects and mathematical reasoning related to series and special functions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant states that the series converges for ##c > 1## and questions if there is an expression for the sum's value as a function of ##c##.
  • Another participant mentions that for some values of ##c##, there are analytic expressions, specifically referring to the Riemann zeta function.
  • A different participant argues that there is no known closed expression for ##\sum_{n=1}^\infty \frac{1}{n^3}##, introducing Apéry's constant as a related concept.
  • A later reply reiterates that the Riemann zeta function applies for all ##c > 1##, suggesting a consensus on this point but not on the existence of closed expressions for specific values.

Areas of Agreement / Disagreement

Participants express differing views on whether a closed expression exists for the sum, with some asserting the Riemann zeta function applies while others highlight the lack of closed forms for certain values like ##c = 3##. The discussion remains unresolved regarding the existence of a general expression for all ##c > 1##.

Contextual Notes

There are limitations regarding the known values of the series and the conditions under which the Riemann zeta function applies. The discussion also touches on specific constants like Apéry's constant without resolving the implications of these references.

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TL;DR
sum ##\frac{1}{n^c}## where ##c\gt 1##
##\sum_n \frac{1}{n^c}## converges for ##c\gt 1##. Is there an expression for the value of the sum as a function of ##c##?
 
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mfb said:
For some values there are analytic expressions. It's the Riemann zeta function.
I should have known! It is the zeta function for all ##c\gt 1##.
 

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