Summation of Series with Exponential Terms: Seeking Analytical Expression

[yassermp]

Has anyone heard about a way to find the sum of a serie of this form:
s=$$\sum_i{\exp(a+b\sqrt(i))}$$

 

[tiny-tim]

Has anyone heard about a way to find the sum of a serie of this form:
s=$$\sum_i{\exp(a+b\sqrt(i))}$$

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 = $$e^a\,\sum_n{e^{b\sqrt{n}}$$.

 

[yassermp]

Hi Tiny-tim

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

 

[tiny-tim]

… 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

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! )