Posted by Sean on Sunday, March 15, 2009 at 11:34pm.
If there is a recursively defined sequence such that
a_{1} = sqrt(2)
a_{n + 1} = sqrt(2 + a_{n})
Show that a_{n} < 2 for all n ≥ 1

Math  Reiny, Sunday, March 15, 2009 at 11:54pm
a1 = √2
a2 = √(2 + √2)
a3 = √(2 + √(2 + √2))
I see a pattern
an = √( 2 + √(2 + ...
let x = √( 2 + √(2 + ...
square both sides, that will drop the leftmost √ on the right side
x^2 = 2 + √( 2 + √(2 + ...
x^2  2 = √( 2 + √(2 + ...
but the right side is what I originally defined as x
so
x^2  2 = x
x^2  x  2 = 0
(x2)(x+1) = 0
x = 2 or x=1
(clearly x = 1 is an extraneous solution)
so an must have a limit of 2.
and since n is a finite number, term a_{n} < 2 .
an interesting loop on the calculator is this
1) enter √2
2) press =
3) plus 2
4) press =
5) press √
6) repeat step 2)
You should get appr 2 correct to 5 decimals after about 10 iterations. 
Math  Sean, Monday, March 16, 2009 at 12:52am
You prove that the sequences converges to 2, but you don't necessarily prove that it will never exceed 2...
Thanks for the great help though 
Math  Reiny, Monday, March 16, 2009 at 12:58am
I proved that its limit is 2
i.e. it will actually never reach 2, only if I take all of its infinite terms.
so clearly it can never exceed 2 if it can't reach 2. 
Math  Sean, Monday, March 16, 2009 at 1:37am
You are absolutely correct. Thanks so much for the help on this!
I'd love to pay you guys back for this service :) 
Math  Count Iblis, Monday, March 16, 2009 at 10:40am
The only thing missing here is that you must show that the series is monotonously increasing, i.e. that
a_{n+1} > a_{n} (that should be easy). This, combined with the fact that the limit is 2 proves that a_n < 2.
If you have a sequence that is not monotonously increasing (or decreasing), it isn't necesarily true that the limiting value isn't attained for finite n. Take e.g. the sequence
sin(pi n/7)/n
which tends to zero, but it is also equal to zero for n a multiple of 7. 
Math  mahlon, Monday, March 16, 2009 at 7:00pm
Brianna arrived at county fair at 10:25. she left at 11:40. how long was she at the fair?