Infinite product with non homogenous recurrence relation

Question

Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ Define $$P_n=\prod_{i=1}^n(1+\frac1{a_i})$$ Find $P_n$ as n approaches $\infty$

I'm not sure where to start tbh. I don't know how to solve the recurrence relation because of the '$n$'

Any help is appreciated!

Answer 1

Hint: $\displaystyle\; 1+\frac{1}{a_k}=\frac{1+a_k}{a_k}=\frac{a_{k+1}}{(k+1) a_k}\,$, then the product telescopes to:

$$\require{cancel} \prod_{k=1}^n \left(1+\frac{1}{a_k}\right) = \frac{\cancel{a_2}}{2 a_1} \cdot \frac{\bcancel{a_3}}{3 \cancel{a_2}} \cdot \ldots\ \frac{a_{n+1}}{(n+1)\bcancel{a_n}} = \frac{a_{n+1}}{(n+1)!} $$

Also, the original recursion can be written as $\,\displaystyle \frac{a_n}{n!} = \frac{a_{n-1}}{(n-1)!}+\frac{1}{(n-1)!}=\ldots\,$.