# How can I reduce this?

1. Nov 16, 2009

### theneedtoknow

1. The problem statement, all variables and given/known data
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

2. Relevant equations

3. The attempt at a solution

2. Nov 16, 2009

### tiny-tim

Hi theneedtoknow!

(try using the X2 tag just above the Reply box )
oooh, you have made it difficult!

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