I am trying to prove the Post-Widder Inversion formula $$ f(t) = \lim_{n \to \infty} \frac{(-1)^n}{n!} \left(\frac{n}{t}\right)^{n + 1} \hat f^{(n)}\left( \frac{n}{t} \right), $$ for $f(t) = T(t)x$ with generator $A$ and the Laplace transform $\hat{f}$. The exponent $\cdot^{(n)}$ denotes the $n$th derivative. I want to use the exponential formula $$ T(t)x = e^{tA}x = \lim_{n \to \infty} \left[ \frac{n}{t} \left(\frac{n}{t} - A\right)^{-1}\right]^n x \tag{1}$$ but somehow I am missing a power of $n$. This is my proof so far:
I know that $$ \hat f^{(n)}(u) = (-1)^n \int_0^\infty s^n f(s) e^{-us} \;ds. $$ Now partial integration gives me $$ \int_0^\infty s^n f(s) e^{-\frac{n}{t} s} ds = n! \left(\frac{t}{n}\right)^n \int_0^\infty f(s) e^{-\frac{n}{t}s} ds = n! \left(\frac{t}{n}\right)^n \left(\frac{n}{t} - A\right)^{-1}. $$
This equality yields $$ \frac{(-1)^n}{n!} \left(\frac{n}{t}\right)^{n + 1} \hat f^{(n)}\left( \frac{n}{t} \right) = \left(\frac{n}{t}\right)\left(\frac{n}{t} - A\right)^{-1} \tag{ 2}, $$ where I used the fact that $$ (\lambda - A)^{-1} x = \int_0^\infty e^{-\lambda t} T(t) x dt. $$ As you can see, I am missing a power of $n$ in (2) if I want to apply (1).
Where did I make a mistake?
Found the error: the partial integration I performed is wrong, but I have no idea on how to correct this error. Any hints?
EDIT 2: Note that I am not looking to "just" prove the inversion formula. I am interested in a proof making use of the exponential formula $(1)$ and, if possible, a way to "fix" my proof.
The proof of Post's inversion formula is already in e.g. http://www.rose-hulman.edu/~bryan/invlap.pdf