# Inverse Laplace Transform of

1. Jul 9, 2005

### phoenixthoth

I'm looking to find the ilt of $$\frac{1}{s(e^s-1)}-\frac{1}{s^2(e^s-1)}$$.

I realize that I can combine this into a rational function whose ilt is $$1-t$$ or something like that and so the main part I'm interested in is $$\frac{1}{e^s-1}$$. I'm not finding it in any table except when I expand it as a geometric series; then it's a sum of delta functions or something but I was wondering if there was a better closed form. Once I get the ilt of the rational expression and the inverted exponential, I can convulte the two, right?

Any help appreciated.

Oh, btw, what I'm trying to do is find the asymptotic behavior of $$\sum_{k=1}^{t-1}(1-t/k)^k$$. I took the lt of this and I *think* the first two terms in its Laurant series expansion are given above, the third term being really small. I'm then hoping that the ilt of the first two terms in the lt will give me the asymptotic behavior of the original.

Thanks.