My sophomore year of college I went through Euler's derivation of the Gamma Function in a translated version of his Opera Omnia. The proof involved division by zero, along with other liberties, and would not be acceptable by modern standards. However, as the end result turned out to be correct, I felt there had to be a way to justify some of the liberties he took and put it in a form that would be acceptable by modern standards. Recently I went through and did this. The only step I'm not completely certain of is this last one. I was hoping someone would take the time to verify it and point out any errors I may have made. Here it is:
I've shown that given a natural number n, for every $k \in \mathbb{N}$
$$ \frac{n!k^n}{(k+2)(k+3)\cdots(k+n+1)} = \int_0^1 \left(k(1-x)^{1/k+1}\right)^n dx $$
Assuming for the moment that the sequence of integrands is dominated by an integrable function we can take the limit of both sides and move the limit under the integral to get $$ n! = \int_{[0,1]} \lim_{k \to \infty} (k(1-x)^{1/k+1})^n d\mu = \int_{[0,1]}\bigg(\lim_{k \to \infty} k(1-x)^{1/k+1}\bigg)^n d\mu$$
By L'hopital, $$ \lim_{g \to 0} \frac{1-x^{\space g/1+g}}{g} = - ln(x)$$ and hence by the sequential version of limits with the sequence $\frac{1}{k}$
$$ k (1-x)^{1/k+1} \rightarrow -ln(x)$$
giving $$ n! = \int_{[0,1]} (-ln(x))^n$$. The familiar version is then obtained by the substitution $x=e^{-t}$ As this is absolutely Riemann integrable it is lebesgue integrable, and since the sequence of integrands are increasing they are bounded by their limit. This then justifies our use of the dominated convergence theorem. Does anything here appear incorrect?