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

In summary: The polylogarithm can be useful in some cases. Thank you for pointing that out.Ah yes, of course. The polylogarithm can be useful in some cases.
  • #1
albertshx
13
0
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!
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
The analytical expression of this series is the Lerch's function.
 
  • #6
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.
 
  • #7
Thank you all! I get the idea!
 
  • #8
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.
 
  • #9
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.
 

1. What is a partial sum in a series?

A partial sum in a series is the sum of a finite number of terms in the series. It is used to approximate the value of an infinite series by adding a finite number of terms.

2. How do I find a function to approximate the partial sum of a series?

In order to find a function to approximate the partial sum of a series, you can use various methods such as Taylor series, Maclaurin series, or power series. These methods involve finding a polynomial function that closely matches the given series.

3. What is the purpose of approximating the partial sum of a series?

The purpose of approximating the partial sum of a series is to find an estimate of the value of an infinite series without having to add an infinite number of terms. This can be useful in situations where the exact value of the series is not known or is too difficult to calculate.

4. Can a function perfectly approximate the partial sum of a series?

No, a function can only approximate the partial sum of a series. This is because an infinite series has an infinite number of terms, and a function can only have a finite number of terms. The approximation will get closer to the true value of the series as more terms are added to the function.

5. Are there any limitations to using a function to approximate the partial sum of a series?

Yes, there are limitations to using a function to approximate the partial sum of a series. The function may only be accurate within a certain range of values and may not be able to provide an accurate estimate for all values of the series. Additionally, the function may become more complex as more terms are added to improve the approximation.

Similar threads

Replies
3
Views
2K
Replies
3
Views
1K
Replies
7
Views
1K
Replies
2
Views
656
Replies
1
Views
793
Replies
2
Views
829
Replies
3
Views
2K
Replies
11
Views
1K
Replies
2
Views
1K
Back
Top