Largest prime factor

[Tolaso J Kos]

Find the largest prime factor of: $$\frac{1}{\sum \limits_{n=1}^\infty \frac{2012}{n(n+1)(n+2)(n+3)\dots(n+2012)}}$$

[Demetres]

Note that \[ \frac{2012}{k(k+1)\cdots (k+2012)} = \frac{1}{k(k+1)\cdots(k+2011)} - \frac{1}{(k+1)(k+2)\cdots (k+2012)}.\] So the sum \[ \sum_{n=1}^{\infty} \frac{2012}{n(n+1)\cdots (n+2012)}\] is telescopic and equals \(1/(2012)!\). In particular the expression of the question is equal to \(2012!\) and so it is indeed an integer. (The question wouldn't make sense otherwise!) Its largest prime factor is the largest prime in the set \(\{1,2,\ldots,2012\}\). It can be checked that \(2011\) is prime. and so it is the largest prime factor of the expression.

[Tolaso J Kos]

A relative topic can be found in the link: Series.
The topic is about the evaluation of the series, which indeed telescopes. But I have another way in mind too, with the use of Beta & Gamma functions. Demetres thank you for the solution. It comes from Putnam competition and I think it was quite recent.