I'm trying to figure out the Asymptotic Approximation for the Laplace Transform but I am stuck at finding the order of the Remainder.
The transform is: $F(x) = \int_0^{+\infty}e^{-xt}f(t){\rm d}t, x>0$
Using repeated integration by parts, I showed that: $$F(x) = \sum_{n=0}^N \frac{f^{(n)}(0)}{x^{n+1}} + R_N(x)$$
where: $$R_N(x) = \frac{1}{x^{N+1}}\int_0^{\infty}e^{-xt}f^{(N+1)}(t){\rm d}t$$
Now I need to show that: $R_N(x) = O(x^{-(N+2)})$ as $x \rightarrow \infty$. I am trying to use the formula from my notes:
$$|R_N(x)| \leq \frac{|x|^{N+1}}{(N+1)!}\sup{|f^{(N+1)}(t)|}$$
but to be honest, I don't really understand what happens in this formula and especially how to apply here.
Any ideas?
Many thanks!