What Is the Correct Partition Function for a Spin System?

[happyparticle]

Homework Statement

Consider a spin i, integer or half integer with the states n = -i, -(i-1),..(i-1), i
Z component of the spin is $S_z = n\hbar$ and the energy eigenvalues of this system in a magnetic field are given by: $E_n = nh$
Find the partition function in term of 2 sinh ratio

Relevant Equations

$Z = \sum_{-i}^{i} = e^{-E_n \beta}$

$Z = \sum_{-i}^{i} = e^{-E_n \beta}$

$Z = \sum_{0}^j e^{nh\beta} + \sum_{0}^j e^{-nh\beta}$
Those sums are 2 finites geometric series
$Z = \frac{1- e^{h\beta(i+1)}}{1-e^{h\beta}} + \frac{1-e^{-h\beta(i+1)}}{1-e^{-h\beta}}$
I don't think this is ring since from that I can't get 2 sinh. However, I'm not sure where is my error.

 

[vanhees71]

Note that you've included $n=0$ twice!

 

[vela]

I don't think this is ring since from that I can't get 2 sinh. However, I'm not sure where is my error.

This might help: $1-e^x = e^{x/2}(e^{-x/2}-e^{x/2})$.

 

[happyparticle]

Thank you! I didn't see that I had included n=0 twice.
I spent hours trying to figure out what was wrong.