How can I simplify this thermodynamics problem?

[theneedtoknow]

Homework Statement

This is actually from a thermodynamics problem, but to just reduce it to math, I've gotten rid of a bunch of physical constants and replaced a larger term with simply "a" , but basically I have to show that:
a^2 * e^a / (e^a - 1)^2
reduces to 1 - (a^2)/12 as a goes to 0
i'm suppsed to approximate the exponential and keep terms up until a^3, until the final answer where I can throw out anything smaller than a^2
so e^a = 1 + a + a^2/2 + a^3/6

after expanding, the denominator is (a^2 + a^3 + 7/12 *a^4 + a^5/6 + a^6/36)
factoring out an a^2 and cancelin with the a^2 in the numerator i get

(1 + a + a^2/2 + a^3 / 6) / (1 + a + 7/12*a^2 + a^3/6 + a^4/36)

I don't really know how to proceed from here, even if I assume it is at this point where i can throw out the last 2 terms of the denominator and the last term of the numerator...I just don't see how i can get 1 - (a^2)/12

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

Hi theneedtoknow!

(try using the X² tag just above the Reply box )

I have to show that:
a^2 * e^a / (e^a - 1)^2
reduces to 1 - (a^2)/12 as a goes to 0

oooh, you have made it difficult!

Hint: e^a/(e^a - 1)² = 1/(e^a/2 - e^-a/2)²