Prove that the sequence $a_{1}=2$ and $a_{n+1} = \frac{a_{n}}{2} + \frac{1}{a_{n}}$ for $n \geq 1$ is bounded, monotone, and find it's limit.

Question

For the sequence:

$a_{1}=2$ and $a_{n+1} = \frac{a_{n}}{2} + \frac{1}{a_{n}}$ for $n \geq 1$

Prove that that the sequence is bounded.
Prove that the sequence is monotone.
Find the limit of the sequence.

I can do this for basic recursive sequence definitions but the presence of two fractions with sequence definitions in each one has me a bit stumped.

Use induction to prove everything (use a candidate for it limit to prove boundedness.) — Masacroso, Nov 27 '16 at 22:54
Hint: solve $L=\frac{L}{2}+\frac{1}{L}$, then consider the sequence $\epsilon_n = a_n-L$. (There are two values of $L$; can you see which one is most useful?) — J.G., Nov 27 '16 at 23:01
Closely related: Prove that the sequence $a_{n+1} =\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right)$ is convergent and find its limit. — dxiv, Nov 27 '16 at 23:21
Check this question closely related too: http://math.stackexchange.com/questions/1977236/for-a0-and-x-0-ge-a-prove-that-the-sequence-defined-as-x-n1-x-na-x-n — Masacroso, Nov 27 '16 at 23:28

score 0 · Answer 1 · answered Nov 27 '16 at 22:54

0

Hints: You can do by induction that $1\leq a_n \leq 2$ for all $n$. Then you can use this fact that the sequence is monotone. For the limit, just do the usual thing.

answered Nov 27 '16 at 22:54

Isko10986

306

Prove that the sequence $a_{1}=2$ and $a_{n+1} = \frac{a_{n}}{2} + \frac{1}{a_{n}}$ for $n \geq 1$ is bounded, monotone, and find it's limit.

1 Answers1