What is the Limit of the Partition Function in the Low Temperature Regime?

[quanlop93]

Homework Statement

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 $$kt \gg \in\\ kt \ll \in$$

Homework Equations

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?

 

[Oxvillian]

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

 

[DrClaude]

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

 

[quanlop93]

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

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.

 

[DrClaude]

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

 

[quanlop93]

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

Got it now. Thank you.