TITLE: A limit of a sequence statistifying $S_{n} = \frac{1}{2}(a_{n}+\frac{1}{a_{n}})=a_1+a_2+...+a_n$
QUESTION [5 upvotes]: Sequence $\{a_n\}$ is a positive sequence and satisfies $S_{n} = \frac{1}{2}(a_{n}+\frac{1}{a_{n}})$ where $S_n = a_1+a_2+...+a_n$. Find $\lim_{n\to \infty} S_{n+1}*(S_{n}-S_{n-1})$

REPLY [6 votes]: Hint:
 since $a_{1}=1$
$$a_{n}=S_{n}-S_{n-1}$$
then we have
$$2S_{n}=S_{n}-S_{n-1}+\dfrac{1}{S_{n}-S_{n-1}}\Longrightarrow S^2_{n}-S^2_{n-1}=1$$
so
$$S_{n}=\sqrt{n},$$
so
$$S_{n+1}(S_{n}-S_{n-1})=\sqrt{n+1}(\sqrt{n}-\sqrt{n-1})\to \dfrac{1}{2}$$