Thread: 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?

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:



squaring:



rearranging:



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

CB