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September 24th, 2008, 11:25 AM   #1
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Sequence convergence proof

So i have this problem that asks to prove that the following sequence convergences, and find it its limit.
a1=sqrt (1)
a2= sqrt (1+sqrt1)
a3= sqrt ( 1+sqrt(1+sqrt(1)))
a4= sqrt (1+sqrt(1+sqrt(1+sqrt(1))))
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September 24th, 2008, 12:23 PM   #2
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Re: Sequence convergence proof

First. You show that this sequence is strictly increasing (use induction).

If sqrt 1 < sqrt (1+sqrt 1) means s1<s2, then hypothesize sn < sn+1.

Use induction again to show that it is bounded above. ex. s1 = sqrt 1 < 2.
If sn <2 then sn+1 = sqrt(1+sn) < sqrt (1+2) < 2...
This isn't the tightest form of this, but we've shown that {sn} is increasing and bounded above.

This means that {sn} must converge to some number s. We know that it is less than or equal to whatever
you show it to be bounded by. Be careful about when it could be equal to this upper bound. Note we could have said
that it is bounded above several numbers. This doesn't mean the sequence will converge to it.
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September 24th, 2008, 02:32 PM   #3
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Re: Sequence convergence proof

The above post shows how to prove that the sequence has a limit. To show what the limit is, substitute the expression into itself:

x = sqrt(1 + sqrt(1 + sqrt(1 + ...)))
x = sqrt(1 + x)
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