Proving Limit of Sequence: an+1=an/2 + 1/an

[Harmony]

Homework Statement

a_n+1=a_n/2 + 1/a_n

Prove that the above sequence converge and find the limit.

Homework Equations

The Attempt at a Solution

I have used Maple 12 to compute up to 10 term, using different initial value of a₀. I found that the sequence is approaching square root of two when a₀ is positive, and negative of square root of positive when a₀ is negative. From the software, I know that the sequence is convergent, but it is rather difficult to find a mathematical proof.

Is it reasonable to say that a_n approximate to a_n+1 when n is approaching infinity? If so how can I prove it? Can I substitute both a_n and a_n+1 with a_infinity and find the limit?

Thanks in advanced.

 

[Pere Callahan]

If you know that your thus defined sequence $(a_n)_{n\geq0}$ converges to some number c then the sequence $(a_{n+1})_{n\geq0}$ converges to that same number.
Then you can use the continuity of the function $x\mapsto \frac{x}{2}+\frac{1}{x}$ to take the limit of the equation $a_n=\frac{a_n}{2}+\frac{1}{a_n}$ to obtain $c=\frac{c}{2}+\frac{1}{c}$ which has the solution $c=\sqrt{2}$ of course.

So if the sequence is convergent the limit is $\sqrt{2}$.

But in order to conclude that your sequence converges in the first place you need to work some more. How would you prove in general that a sequence is convergent?

 

[HallsofIvy]

Homework Statement

a_n+1=a_n/2 + 1/a_n

Prove that the above sequence converge and find the limit.

Homework Equations

The Attempt at a Solution

I have used Maple 12 to compute up to 10 term, using different initial value of a₀. I found that the sequence is approaching square root of two when a₀ is positive, and negative of square root of positive when a₀ is negative. From the software, I know that the sequence is convergent, but it is rather difficult to find a mathematical proof.

Is it reasonable to say that a_n approximate to a_n+1 when n is approaching infinity? If so how can I prove it? Can I substitute both a_n and a_n+1 with a_infinity and find the limit?

Thanks in advanced.

Well, you shouldn't call it a_infinity. Rather, IF the sequence converges, say to some number A, then, taking limits on both sides of the equation A= A/2+ 1/A. From that, multiplying on both sides by A, A²+ A²/2+ 1 or (1/2)A²= 1 or A²= 2. Assuming that a₀> 0 then $A= \sqrt{2}$.

That is, of course, IF the sequence converges. With that information, we can then use "monotone convergence" to show that it does, in fact, converge.

 

[statdad]

As a slight (I hope not too far-reaching) expansion on the previous correct ideas:

1. Try to show that the sequence is non-decreasing (i.e. $$a_{n+1} \ge a_n$$)
2. Show that the sequence is bounded above ($$a_n \le M$$ for some
positive number $$M$$)

These two steps together will be enough to prove that the sequence converges; you've already seen how to find the limit.