I'm getting a little stuck on this question. The question is: show that for $KT \gg h\omega$, the first law of Planck: $\displaystyle U =\frac{h\omega}{e^{(h\omega/KT)}-1} \approx KT - \frac{h\omega}{2} + \cdots$ (where ... represents the higher order terms of $\frac{h\omega}{KT}$)
I've come this far, using the Taylor expansion of $\displaystyle e^{\frac{h\omega}{KT}} = 1 + \frac{h\omega}{KT} + \frac{{(hw)}^2}{2{(KT)}^2} + \cdots$
Then $$U = \frac{h\omega}{1 + \frac{h\omega}{KT} + \frac{{(hw)}^2}{2{(KT)}^2} + \cdots} = \frac{KT}{1 + \frac{h\omega}{2KT} + \cdots}$$
I get stuck on that point though. I can see how the top equation is very close to the calculated equation so far, but I can't get the forms to equal each other (especially the minus sign prior to $\frac{h\omega}{2}$). I suspect that the $KT \gg h\omega$ has to do something with it, but I don't know how to simplify my form further. I hope someone can help me out!