Closed Form Solution for Series with Exponential and Power Terms?

  • Thread starter rman144
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In summary, the conversation is about finding a closed form solution for the series \sum_{n=1}^{\infty}(-1)^{n}\frac{e^{-(nx)^2}}{n^{1-m}}, with m being an integer and 0<x<oo. The discussion includes considering a theta-type function as a possible solution and using a Taylor series to simplify the problem. The purpose of finding a closed form solution is to potentially gain more insight and use well-documented properties of known functions for analysis or numerical computations.
  • #1
rman144
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[tex]\sum_{n=1}^{\infty}(-1)^{n}\frac{e^{-(nx)^2}}{n^{1-m}}[/tex]

Where m is an integer and 0<x<oo. I need a closed form solution, and was thinking something along the lines of a theta-type function, but cannot seem to locate any identities that match. Anyone have a suggestion?
 
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  • #2
rman144,

I'm curious what you consider "closed form" and what you will use it for (analysis, numerics, etc.). Clearly, for x>1 this is an excellent representation that converges quickly - I would be surprised if a theta function (or hypergeometric or a G-function or ...) would be any "simpler" in reality or easier to compute, although on paper you may be able to write a small number of special functions, at best. Of course for x<1 the series may leave something to be desired, as I am pretty sure it can have a large number of terms with increasing magnitude before the terms start to decrease.

Just curious.

Jason
 
  • #3
rman144,

I just re-read my post - it sounds like I am questioning the utility of finding another representation of the series. I didn't mean it that way - honest! I'm guessing most of us have been in a similar situation of looking for a nicer representation that may yield more insight, allow us to use well documented properties of known functions, allow us to use code we already have to compute it, etc. Anyway, I really am curious about the source of the series. Also, are there any other constraints on m beyond being an integer (even/odd, positive/negative)?

Regards,

Jason
 
  • #4
Perhaps replace the exponential by its Taylor series, switch the double sums? It's something to try.
 
  • #5
Lol, I took no offense. Thank you for the help.
 

1. What is a series in mathematics?

A series in mathematics is a sum of numbers that follow a specific pattern or sequence. It is an important concept in calculus and is often used to solve problems involving infinite sequences of numbers.

2. How do I find the sum of a series?

To find the sum of a series, you can use a specific formula or method depending on the type of series. For example, for a geometric series, you can use the formula Sn = a(1-r^n)/(1-r), where a is the first term and r is the common ratio. For an arithmetic series, you can use the formula Sn = (n/2)(a+l), where n is the number of terms, a is the first term, and l is the last term.

3. What is the difference between a series and a sequence?

A series is a sum of terms, while a sequence is a list of numbers that follow a specific pattern or rule. In a series, the terms are added together, while in a sequence, the terms are listed in a specific order.

4. How do I know if a series is convergent or divergent?

A series is convergent if the sum of its terms approaches a finite number as the number of terms increases. It is divergent if the sum of its terms approaches infinity as the number of terms increases. To determine if a series is convergent or divergent, you can use various tests such as the comparison test, the ratio test, or the integral test.

5. Can I use a calculator to find the sum of a series?

Yes, you can use a calculator or a computer program to find the sum of a series. However, it is important to understand the concept behind the series and the formula or method used to find the sum in order to verify the accuracy of the calculated result.

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