Bound for S: Sum of n^k e^(-an)

Thread starter bruno67
Start date Dec 5, 2011
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In summary, "Bound for S: Sum of n^k e^(-an)" is a mathematical expression used to approximate the value of certain integrals. It involves finding the sum of a series using methods like the Euler-Maclaurin formula and is important in various fields such as physics and engineering. However, it can only be used for series that follow a specific form and may have limitations in accurately representing the actual value of the series.

Dec 5, 2011

bruno67

I am looking for a bound for the following expression

[tex]S=\sum_{n=1}^N n^k e^{-an}[/tex]
where a>0 and k=1, 2, 3, or 4, apart from the obvious one:

[tex]S\le \frac{n+1}{2} \sum_{n=1}^N e^{-an} = \frac{n+1}{2}
\frac{1-e^{-Na}}{e^a-1}[/tex]

Last edited: Dec 5, 2011

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Dec 5, 2011

bruno67

I got it. S(k) is bound by the integral

[tex]S\le \int_1^{N+1} x^k e^{-ax} dx[/tex]

Related to Bound for S: Sum of n^k e^(-an)

1. What is "Bound for S: Sum of n^k e^(-an)"?

"Bound for S: Sum of n^k e^(-an)" is a mathematical expression that represents the sum of a series where the terms are raised to a power, n, and multiplied by the exponential function, e^(-an). This expression is often used in mathematical analysis and can be used to approximate the value of certain integrals.

2. How is "Bound for S" calculated?

The calculation of "Bound for S" involves finding the sum of the series by adding up all the terms until a certain number, n, is reached. This can be done using various methods such as the Euler-Maclaurin formula or the Euler-Maclaurin summation formula. The resulting value is then used as an upper bound for the actual value of the series.

3. What is the importance of "Bound for S" in mathematics?

"Bound for S" is important in mathematics because it provides a way to approximate the value of certain integrals, which can be difficult to solve exactly. It also has applications in physics, engineering, and other fields where integrals are used to model real-world problems.

4. Can "Bound for S" be used for any series?

No, "Bound for S" can only be used for series that follow the specific form of n^k e^(-an). If the series does not follow this form, then a different method must be used to approximate its value.

5. Are there any limitations to using "Bound for S"?

Yes, there are some limitations to using "Bound for S" as an approximation for the value of a series. It is only an upper bound, so it may not accurately represent the actual value of the series. Additionally, the accuracy of the approximation depends on the value of n used, and for some series, the value of n may need to be very large to get a good approximation.