Limit of partition function

1. Nov 11, 2015

quanlop93

1. The problem statement, all variables and given/known data
Ground state energy is set at 0.
$$E_n=\left(1-\frac{1}{n+1}\right)\in$$ with no degeneracy $$(\Omega(n)=1); (n=0,1,2.....)$$
Write down the partition function and look for its limit when $$kt \gg \in\\ kt \ll \in$$

2. Relevant equations

3. The attempt at a solution
Partition function for this is $$Z=\sum_{n=0}^\infty e^{-\beta\left(1-\frac{1}{n-1}\right)\in}$$
Consider Z when $kt \ll \in$ then $\beta e \gg1$ then $e^{-\beta e} \rightarrow 0$ This leads to the whole summation will go to 0. But we know that at low temperature, Z always goes to 1.
I have tried to calculate the summation but this series is divergent.
How can I change the calculation to reach Z =1?

2. Nov 11, 2015

Oxvillian

I agree - if you have an infinite number of energy levels, bounded above by $\epsilon$, then the partition function diverges.

3. Nov 11, 2015

Staff: Mentor

What happens if you single out $n=0$?

4. Nov 19, 2015

quanlop93

With n = 0 $$\left(1-\frac{1}{0+1}\right)=0$$ then Z=1 in two cases. But its supposed to be 1 just in the case that the temperature is low $$kt\ll\epsilon$$
I have tried some direct methods to find the limit of this function, but it turned out that the function is divergent. Then all of them became useless.

5. Nov 19, 2015

Staff: Mentor

What I meant is take $n=0$ out of the sum in the low T limit, and you recover $Z=1$ as expected.

6. Nov 28, 2015

quanlop93

Got it now. Thank you.