# convergence of a recursive sequence and calculate the limit

how to show that the sequence $x_{n+1}=\frac{x_n}{2}+\frac{5}{x_n}$, $x_1=2$ is convergent.
I tried to prove using induction that it is bounded but couldn't work it out. Only thing i could figure out is that the limit of sequence is $\sqrt{10}$ so it is convergent.

• @Salahamam_Fatima: That is wrong. – Aryabhata Aug 6 '17 at 20:12

By AM>=GM we have that

$$x_{n+1} \ge 2\sqrt{(x_n/2)\cdot(5/x_n)} = \sqrt{10}$$

Now the sequence is

$$x_{n+1} = f(x_n)\quad\text{where}\quad f(x) = \frac{x}{2} + \frac{5}{x}$$

For $x \ge \sqrt{10}$ we have that $$|f'(x)| = \left|\frac{x^2 - 10}{2x^2}\right| = \frac{x^2 - 10}{2x^2} \lt \frac{1}{2}$$

Thus the sequence is convergent.

An alternative is to use the fact that $x_{n+1} \gt \sqrt{10}$ (for non constant $x_n$) to show that the sequence is monotonically decreasing (show that $x_{n+1} - x_n \lt 0$) and bounded.