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

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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