Evaluation of $\lim_{n \to \infty} ((n+1)!\ln (a_n))$

Question

Consider the sequence $(a_n)_{n \geq1}$ such that $a_0=2$ and $a_{n-1}-a_n=\frac{n}{(n+1)!}$. Evaluate $$\lim_{n \to \infty} ((n+1)!\ln (a_n))$$

Could someone hint me as how to achieve value of $a_n$ from given information.

Answer 1

Hint. The key word is telescoping sums, $$ a_{n-1}-a_n=\frac{n}{(n+1)!} \implies a_0-a_n=\sum_{k=1}^n\frac{k}{(k+1)!} $$ and $$ a_0-a_n=\sum_{k=1}^n\frac{(k+1)-1}{(k+1)!}=\sum_{k=1}^n\frac1{k!}-\sum_{k=1}^n\frac1{(k+1)!}=1-\frac1{(n+1)!} $$ Can you take it from here?

Answer 2

Hint: Show by induction that $a_n=1+\frac{1}{(n+1)!}$