Evaluate $\lim\limits_{n \to \infty}\left(\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots+\frac{1}{1+a_n}\right).$ Problem
Let $a_1=3,a_{n+1}=a_n^2+a_n(n=1,2,\cdots)$. Evaluate
$$\lim_{n \to \infty}\left(\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots+\frac{1}{1+a_n}\right).$$
Attempt
Notice $$\frac{1}{1+a_n}=\frac{a_n}{a_n+a_n^2}=\frac{a_n}{a_{n+1}}$$
Then
$$\sum_{k=1}^{n}\frac{1}{1+a_k}=\sum_{k=1}^n \frac{a_k}{a_{k+1}}.$$
This will help?

Prove that
$$
\sum_{k=1}^{n} \frac{1}{1 + a_{k}} = \frac{1}{3} - \frac{1}{a_{n+1}}
$$
by induction on $n$. Then the limit is $1/3$. 
