Help me convert Boltzmann distribution/partition function into Geometric series

[asdfTT123]

Homework Statement

3. The following calculaltion shows how the ratio of e to kT affects the
populations of different energy levels. kT is sometimes called the thermal
energy; if it is small relative to e, a particle will not be able to access higher
energy states.
Consider a harmonic oscillator with spacing νe/kT between energy levels, where
ν can take on the values 0, 1, 2, etc.Calculate the fraction of particles in the ν = 3
energy level for e/kT = 4, 1, and 0.2.

Note: The definition of a harmonic oscillator is that the energy levels are equally spaced so the spacing between energy levels is identical.

Homework Equations

The only equations you need for this is the Boltzmann distribution and partition function,

Boltzmann Distribution:
ni/N = e^(-eiB)/z

z = Partition function = summation of e^(-eiB)

B= 1/kT

k = Boltzmann constant = 1.381e-23 J/K

(Sorry for the poor notation but you can look these up online)

The Attempt at a Solution

I know how to do this problem and how to set it up, but the problem here is I don't know how to express the expression as an infinite series.

For example, let's say e/kT = 1:

ni/N = e^(-3 * 1)/(1 + e^(-1) + e^(-2) + e^(-3)...)

My professor says the equation can be expressed in the following geometric series:

1/(1 - e^(-e/kT))

I'm pretty sure that doesn't work because it doesn't result in sensible answers. Does anyone here know how to correctly express it in terms of a series? I realize my explanation is rather hazy but please consult this pdf (problem #3) for more information...you will see if you use his series you will not get the answers he has listed

http://ded.chm.jhu.edu/~pchem/AnswerKeys/Resources/HW Wk13 Ans.pdf

Thanks!

 

[Shooting Star]

Do you know the sum of an infinite GP? (Don't put constants equal to 1; sometimes it becomes difficult afterwards to see exactly where to put it back.)

 

[ogb p]

The Boltzmann distribution and partition function can indeed be expressed as a geometric series. The key is to recognize that the summation in the partition function is a geometric series with a common ratio of e^(-1/kT). This can be written as:

z = 1 + e^(-1/kT) + e^(-2/kT) + e^(-3/kT) + ...

Using the formula for the sum of a geometric series, this can be simplified to:

z = 1/(1 - e^(-1/kT))

Substituting this into the Boltzmann distribution equation, we get:

ni/N = e^(-ei/kT)/(1/(1 - e^(-1/kT)))

Simplifying further, we get:

ni/N = e^(-ei/kT) * (1 - e^(-1/kT))

This can also be written as:

ni/N = e^(-ei/kT) - e^(-ei/kT - 1/kT)

This expression can be used to calculate the fraction of particles in the ν = 3 energy level for different values of e/kT. For example, for e/kT = 4, ni/N = e^(-3*4) - e^(-3*4 - 1/4) = 0.000123.

I hope this helps clarify how the Boltzmann distribution and partition function can be expressed as a geometric series.