# Limit question

• Jan 29th 2010, 05:38 AM
Henryt999
Limit question
Show that this converges and where it converges too.

$a(n+1) = 1/2(an +3/an)$. given that (a1=2)

Well I found out where it converges too by starting with a1=2 that gave me a2 = 7/4, a3 = 97/56 and that is very close too sqrt:3 so that is where it goes. (Thinking).
I am however not sure how to "show" that it converges...?
A general hint on how to show anything here is fine I really wan´t too solve these by myself...
• Jan 29th 2010, 05:50 AM
General
Quote:

Originally Posted by Henryt999
Show that this converges and where it converges too.

$a(n+1) = 1/2(an +3/an)$. given that (a1=2)

Well I found out where it converges too by starting with a1=2 that gave me a2 = 7/4, a3 = 97/56 and that is very close too sqrt:3 so that is where it goes. (Thinking).
I am however not sure how to "show" that it converges...?
A general hint on how to show anything here is fine I really wan´t too solve these by myself...

Do you the mathematical induction?
• Jan 29th 2010, 05:57 AM
Henryt999
Exactly what?
We might have another name for it in europe? "there is something called the inductionaxiom but I am not suppose to use that here..
• Jan 29th 2010, 06:28 AM
Henryt999
Acctually let me refrase the question.
I have to show that it converges. Well I can do that by putting numbers in.
But I can only show that it converges from the left to sqrt3.
Any tips on how I can show that it also converges from the right?
• Jan 29th 2010, 12:58 PM
mr fantastic
Quote:

Originally Posted by Henryt999
Show that this converges and where it converges too.

$a(n+1) = 1/2(an +3/an)$. given that (a1=2)

Well I found out where it converges too by starting with a1=2 that gave me a2 = 7/4, a3 = 97/56 and that is very close too sqrt:3 so that is where it goes. (Thinking).
I am however not sure how to "show" that it converges...?
A general hint on how to show anything here is fine I really wan´t too solve these by myself...

Do you mean

$a_{n+1} = \frac{1}{2} \left(a_n + \frac{3}{a_n}\right)$, given that $a_1=2$ ?

You have found $\sqrt{3}$ as a possible limit. Now you need to use induction to show that $0 < a_{n+1} < a_n$ for all n greater than 1.