# Summation of Series with Exponential Terms: Seeking Analytical Expression

• yassermp
In summary, the conversation discusses a method to sum a series of the form s=\sum_i{\exp(a+b\sqrt(i))} and its application in summing contributions of spherical waves from scattering centers. The conversation also mentions attempting to find an exact analytical expression for this sum using Fourier transform, but concludes that it may be a difficult task.
yassermp
Has anyone heard about a way to find the sum of a serie of this form:
s=$$\sum_i{\exp(a+b\sqrt(i))}$$

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

Hi Tiny-tim

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

yassermp said:
… 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! )

## What is a series summation?

A series summation is a mathematical process of adding up a sequence of numbers in a specific order. It involves finding the sum of a series or adding together all the terms in a series.

## What is the purpose of series summation?

The purpose of series summation is to determine the total value or sum of a series. It is commonly used in mathematics, physics, and engineering to solve problems involving sequences of numbers.

## What are the different types of series summation?

The two main types of series summation are finite and infinite. Finite series have a limited number of terms, while infinite series have an unlimited number of terms.

## How do you perform series summation?

To perform series summation, you need to follow a specific formula depending on the type of series. For finite series, you can use the formula for arithmetic or geometric series. For infinite series, you can use the convergence test, such as the ratio test or integral test, to determine if the series converges or diverges.

## What are some common applications of series summation?

Series summation is commonly used in various fields such as finance, statistics, and computer science. It is also used in physics to calculate the total energy of a system and in calculus to determine the area under a curve.

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