Sequence $a_1=2$ and $a_n=\frac{2}{1+a_{n-1}}$ is convergent or divergent?

Question

Given a sequence with $a_1=2$ and $a_n=\frac{a_1}{1+a_{n-1}}$, I have to check whether this sequence is convergent or divergent.

I have reached the conclusion till now that its subsequence $a_{2n}$ is an increasing sequence and bounded above, while the subsequence $a_{2n-1}$ is a decreasing sequence and bounded below. So, both the subsequences are convergent. But I am not able to find $\lim_{n\to\infty} a_{2n}$ or $\lim_{n\to\infty} a_{2n-1}$. Can you help me to proceed further or maybe find another way to show its convergence or divergence?

Hello :) Is it true, that $a_1$ is in the nominator? Then you can write $a_n=\frac{2}{1+a_{n-1}}$. — Jochen, Jul 28 '21 at 11:26

score 2 · Accepted Answer · answered Jul 28 '21 at 11:31

2

If $a_{2n} \to a$ and $a_{2n-1} \to b$ then $a=\frac 2 {1+b}$ and $b=\frac 2 {1+a}$. Using these two equations show that $a=b$

answered Jul 28 '21 at 11:31

Kavi Rama Murthy

311,013

Sequence $a_1=2$ and $a_n=\frac{2}{1+a_{n-1}}$ is convergent or divergent?

1 Answers1