yassermp
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Has anyone heard about a way to find the sum of a serie of this form:
s=[tex]\sum_i{\exp(a+b\sqrt(i))}[/tex]
s=[tex]\sum_i{\exp(a+b\sqrt(i))}[/tex]
Summation of Series with Exponential Terms: Seeking Analytical Expression
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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 [tex]r_j=\sqrt{y^2+(z-z_j)^2}[/tex], with [tex]z_j=jh[/tex], j=1...n. The original sum is:
s=[tex]\sum_j{e^{ikr_j}/r_j[/tex]
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 = [tex]e^a\,\sum_n{e^{b\sqrt{n}}[/tex].
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 [tex]r_j=\sqrt{y^2+(z-z_j)^2}[/tex], with [tex]z_j=jh[/tex], j=1...n. The original sum is:
s=[tex]\sum_j{e^{ikr_j}/r_j[/tex]
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
