- #1
yassermp
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Has anyone heard about a way to find the sum of a serie of this form:
s=\sum_i{\exp(a+b\sqrt(i))}
s=\sum_i{\exp(a+b\sqrt(i))}
Summation of Series with Exponential Terms: Seeking Analytical Expression
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- Thread starter yassermp
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Discussion Overview
The discussion revolves around finding an analytical expression for the sum of a series of the form \( s = \sum_i{\exp(a+b\sqrt{i})} \). Participants explore the mathematical properties and potential applications of this series, particularly in the context of scattering contributions from spherical waves.
Discussion Character
- Exploratory
- Technical explanation
- Mathematical reasoning
Main Points Raised
- One participant inquires about the purpose of summing the series and notes that the term "a" can be factored out of the summation.
- Another participant explains that the series arises in the context of summing contributions from spherical waves, providing a specific formulation involving \( r_j = \sqrt{y^2+(z-z_j)^2} \).
- A participant expresses a desire for an exact analytical expression for the sum, acknowledging the complexity of the problem and mentioning attempts with Fourier transforms that seem unproductive.
- One participant advises against using "i" as an index due to potential confusion with complex numbers.
Areas of Agreement / Disagreement
Participants do not reach a consensus on how to approach the problem or whether an analytical expression can be found. The discussion remains unresolved with multiple perspectives presented.
Contextual Notes
There are limitations regarding the assumptions made about the series and the potential complexity of the mathematical techniques discussed, such as Fourier transforms. The use of "i" as an index is noted as a potential source of confusion.
- #2
tiny-tim
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yassermp said:
Hi yassermp! Welcome to PF!
Why would you want to sum such a series?
Have you noticed you can take the "a" outside the ∑, and write it:
s = e^a\,\sum_n{e^{b\sqrt{n}}.
- #3
yassermp
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Hi Tiny-tim
Hi Tiny Tim, i see what you say, you are totally right. Essencially, that kind of sum arises when you try to sum contributions of several spherical waves, from scattering centers located at r_j=\sqrt{y^2+(z-z_j)^2}, with z_j=jh, j=1...n. The original sum is:
s=\sum_j{e^{ikr_j}/r_j
Very often some approximations are used here, but i would like to obtain an exact analytical expression (no matter what complicated it could be). Id really thank any usefull sugestion(I know this is not an easy one). I tried a bit with some Fourier transform but i think it takes to an endless road.
Thks
tiny-tim said:Hi yassermp! Welcome to PF!
Why would you want to sum such a series?
Have you noticed you can take the "a" outside the ∑, and write it:
s = e^a\,\sum_n{e^{b\sqrt{n}}.
Hi Tiny Tim, i see what you say, you are totally right. Essencially, that kind of sum arises when you try to sum contributions of several spherical waves, from scattering centers located at r_j=\sqrt{y^2+(z-z_j)^2}, with z_j=jh, j=1...n. The original sum is:
s=\sum_j{e^{ikr_j}/r_j
Very often some approximations are used here, but i would like to obtain an exact analytical expression (no matter what complicated it could be). Id really thank any usefull sugestion(I know this is not an easy one). I tried a bit with some Fourier transform but i think it takes to an endless road.
Thks
- #4
tiny-tim
Science Advisor
Homework Helper
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yassermp said:
Hi yassermp!
Sorry … but I can't help you there.
(btw, not a good idea to use i as an index when you're dealing with complex numbers!
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