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August 31, 2016
Posted by **Sean** on Sunday, March 15, 2009 at 11:34pm.

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Show that a

- Math -
**Reiny**, Sunday, March 15, 2009 at 11:54pma1 = √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 left-most √ 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

(x-2)(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 12:52amYou 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 12:58amI 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 1:37amYou 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 10:40amThe 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 7:00pmBrianna arrived at county fair at 10:25. she left at 11:40. how long was she at the fair?