Please help find a function to approximate the partial sum of this series

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Discussion Overview

The discussion revolves around finding a continuous function to approximate the partial sum of the series defined by a(k) = k^(-s)e^(-tk), where s and t are positive parameters. Participants explore various methods for approximation, including integral transforms and interpolation techniques.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant seeks a differentiable function f(a, s, t) to approximate the infinite sum of the series, specifically for cases where s >= 2 and t > 0.
  • Another participant suggests using integral transforms or interpolation methods, mentioning splines and wavelet analysis as potential approaches.
  • Some participants note that the analytical expression of the series can be related to the Lerch's function and the polylogarithm, with distinctions made between cases based on the parameter a.
  • It is mentioned that the series can be expressed in terms of the polylogarithm when considering the complete series (a=1), while the Lerch function applies to more general cases.

Areas of Agreement / Disagreement

Participants express differing views on the best approach to approximate the series, with some proposing integral transforms and others emphasizing the relationship to specific mathematical functions like the Lerch's function and polylogarithm. No consensus is reached on a single method or function for approximation.

Contextual Notes

Participants acknowledge the complexity of the problem, noting that the effectiveness of different approximation methods may depend on specific conditions such as the boundedness of the domain and the number of polynomial terms used.

albertshx
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Hello every one, now I'm dealing with a series
a(k) = k^(-s)e^(-tk), s,t > 0
I want to find a continuous function to approximate the partial sum, S(n)of it.
I hope there can be a good approximation. Please help me find it, thanks!
 
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Albertshx said:
Hello every one, now I'm dealing with a series
a(k) = k^(-s)e^(-tk), s,t > 0
I want to find a continuous function to approximate the partial sum, S(n)of it.
I hope there can be a good approximation. Please help me find it, thanks!

Ideas that spring to mind include using some kind of integral transform, or an interpolation (or approximation scheme).

With regards to interpolation, splines stick out.

Its really hard to be specific in helping you. If you are looking at approximating a function over the whole real line, look at wavelet analysis. If the domain is bounded, certain polynomials might be better. If you have constraints on how many terms in your polynomial you have then again, it depends.
 
Sorry, even I myself was confused. Let me clarify a bit. I want to find a function f(a, s,t) which is differentiable to s and t to approximate the infinite sum of this series, \sum_{k=a}^{inf}k^{-s}e^{-kt}
in case s>=2, t > 0. Surely the RHS < inf. I have tried integration, but the effect is not satisfactory.
 
chiro said:
Ideas that spring to mind include using some kind of integral transform, or an interpolation (or approximation scheme).

With regards to interpolation, splines stick out.

Its really hard to be specific in helping you. If you are looking at approximating a function over the whole real line, look at wavelet analysis. If the domain is bounded, certain polynomials might be better. If you have constraints on how many terms in your polynomial you have then again, it depends.

Please see my reply. At least, I want an approximation f(s,t) when a = 1.
 
The analytical expression of this series is the Lerch's function.
 
JJacquelin said:
The analytical expression of this series is the Lerch's function.

Actually, if I'm correct, this series can be expressed in terms of the polylogarithm.
 
Thank you all! I get the idea!
 
Actually, if I'm correct, this series can be expressed in terms of the polylogarithm.
In fact, The Lerch Function is more general than polylogarithm. Both are related in some particular cases.
The series considered here can be expessed in terms of polylogarithm in case of complete series (particular case a=1).
In the general case (if parameter a is any integer > 0 ), the series can be expessed in terms of Lech function.
 
JJacquelin said:
In fact, The Lerch Function is more general than polylogarithm. Both are related in some particular cases.
The series considered here can be expessed in terms of polylogarithm in case of complete series (particular case a=1).
In the general case (if parameter a is any integer > 0 ), the series can be expessed in terms of Lech function.

Ah yes, of course.
 

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