Problem: Suppose 0 < sn < 2 and sn+1 = root (sn + 2) for n in N. Prove(adsbygoogle = window.adsbygoogle || []).push({});

0 < sn < sn+1 < 2 holds for all n in N. Does sn converge? If so, what is the limit.

I am able to show that sn+1 < 2 by squaring the equation sn+1 = root (sn + 2) and making a substitution.

How would I go about showing that sn < sn+1?

Also, if sn < sn+1 < 2 for all n, then the series sn must converge (because it is bounded). In order to find the limit, could I take the limit of both sides of the equation (sn+1)^2 = sn + 2?

i.e. lim (sn+1)^2 = lim (sn+2)).

Let s = lim sn = lim sn+1

Then s^2 = s + 2

s^2 - s - 2 = 0

But then s = 2 and s = -1, which would indicate that there is no limit.

So which is it - is there a limit or not? (I thought that if a sequence is bounded, it must have a limit. Is that correct?)

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# Homework Help: Test Review 3 - convergence of series

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