# infinite sequence

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• Dec 6th 2011, 10:01 PM
jaqueh
infinite sequence
What does the series converge to where
[a_1]=√12
[a_n+1]=√(12+a_n) ?

I tried:
L = √(12+√(12+√(12+...)))
then L^2 = 12+√(12+√(12+...))
then L^∞ = 12^∞ + 12^∞-1...+12
thus L = ∞√(12^∞ + 12^∞-1...+12) but i dont think that this is right.

Then someone on physics forum told me that a_n can be written as √(12+a_n+1) so a_n+1 converges because it is in the series itself, but what is the limit?
• Dec 6th 2011, 10:49 PM
CaptainBlack
Re: infinite series
Quote:

Originally Posted by jaqueh
What does the series converge to where
[a_1]=√12
[a_n+1]=√(12+a_n) ?

I tried:
L = √(12+√(12+√(12+...)))
then L^2 = 12+√(12+√(12+...))
then L^∞ = 12^∞ + 12^∞-1...+12
thus L = ∞√(12^∞ + 12^∞-1...+12) but i dont think that this is right.

Then someone on physics forum told me that a_n can be written as √(12+a_n+1) so a_n+1 converges because it is in the series itself, but what is the limit?

If this converges the limit should satisfy:

$\displaystyle L=\sqrt{12+L}$

squaring:

$\displaystyle L^2=12+L$

rearranging:

$\displaystyle L^2-L-12=0$

This has exactly one positive root and it is the limit you seek.

CB