i was watching a video on extending the factorial and got confused about one step in particular. In 22:45 he claims the following:
$\frac{d}{dx} \left( \lim_{N \to \infty} \sum_{k=1}^{N} \left( \ln{\frac{k}{x+k}} \right) +x \cdot \ln N \right)=\lim_{N \to \infty} \left( \sum_{k=1}^{N} \left(\frac{d}{dx}\left( \ln{\frac{k}{x+k}} \right)\right) +\frac{d}{dx} \left( x \cdot \ln N \right) \right)$
To be able to swap derivative and the limit, the sequence has to be uniformly convergent. From a post i posted earlier, it appears not to be the case. Am i missing something?
My thoughts:
- Is this step actually legal?
- Is there any paper that derives the factorial as seen in video?
Thank you for your answers!