Please see the attached file.

For a CO molecule, the constant ? is approximately 0.00024 eV. Calculate the rotational partition function for a CO molecule at room temperature (300 K), first using the exact forumula:

(j = 0 to ∞)
Zrot = Σ(2j + 1)e-E(j)/kT = Σ(2j + 1)e-j(j + 1)?/kT,

and then using the approximate formula:
(integral is from 0 to ∞)
Zrot ≈ ∫(2j + 1)e-j(j + 1)?/kT dj = kT/?.

The constant
The Boltzmann constant
The temperature is
So we define

Here is the plot of (j, Z) till j is larger than 45. The summation up to yields the same Z of 108, (the difference is less than 1%).
The results are listed in the attached excel file.
So using the summation,

Using the approximate formula: