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## Homework Statement

"Derive a more accurate approximation for the heat capacity at high temperatures, by keeping terms through [itex]x^{3}[/itex] in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than [itex](\frac{ε}{kT})^{2}[/itex] in the final answer. When the smoke clears, you should find [itex]C = Nk(1 - \frac{1}{12}(\frac{ε}{kT})^{2})[/itex]"

## Homework Equations

In addition to the above:

The "exact" formula for the heat capacity was found in an earlier part to the question:

[itex]C = \frac{Nε^{2}e^{\frac{ε}{kT}}}{kT^{2}(e^{\frac{ε}{kT}} - 1)^{2}}[/itex]

## The Attempt at a Solution

I used the Power Series expansion for small x:

[itex]e^{x} ≈ 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6}[/itex]

where [itex]x = \frac{ε}{kT}[/itex]

I expanded all this out, factoring out [itex]x^{2}[/itex] on bottom to cancel the same on top, removed powers of x greater than 2:

[itex]C = N k \frac{x^{2}(1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6})}{(x + \frac{x^{2}}{2} + \frac{x^{3}}{6})^{2}}[/itex]

[itex]C = N k \frac{x^{2}(1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6})}{x^{2} + x^{3} + \frac{7 x^{4}}{12} + \frac{x^{5}}{6} + \frac{x^{6}}{36}}[/itex]

[itex]C = Nk\frac{1 + x + \frac{x^{2}}{2}}{1 + x + \frac{7 x^{2}}{12}}[/itex]

I'm not sure how to take this any further. I have gotten something that almost resembles what I want:

[itex]C (1 + x + \frac{7x^{2}}{12}) = N k (1 + x +\frac{7x^{2}}{12} - \frac{x^{2}}{12})[/itex]

But clearly, I can't just divide by the part attached to C and call it a day. Any tips to help get to the target answer?