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Finding the limit of a sequence.

  • #1

Homework Statement



A sequence [itex]\{s_n\}[/itex] is defined by [itex]s_{n+1} = \frac{1}{2} (s_n + s_{n-1}); s_1 > s_2 > 0[/itex] I have to prove that the sequence is convergent and I have to find the limit.

Homework Equations





The Attempt at a Solution


I tried equating the limit of both sides to get s = (1/2)(s + s) but then I just get s = s. I managed to find that the sequence is bounded between [itex]s_1[/itex] and [itex]s_2[/itex] but the sequence is neither increasing nor decreasing.

For proving the convergence I tried used the Cauchy convergence test.
[itex] |\frac{1}{2}(s_n + s_{n-1}) - s_n| = |\frac{1}{2}(s_{n-1} - s_n)| < \in [/itex] is as far as I got. I can't assume that the s_n tends to a limit s because I haven't proved it does yet. So I'm stuck.

I'm also completely stuck on how to prove the convergence of a sequence defined by a recurrence relation in which the relation involves TWO of the previous terms.

where [itex]s_n = f(s_n, s_{n-1})[/itex].
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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[itex](s_n+ s_{n-1})/2[/itex] is exactly half way between [itex]s_n[/itex] and [itex]s_{n-1}[/itex] That means that the distance between two consecutive terms is half the distance between the preceding termS: [itex]|s_n- s_{n-1}|= (1/2)|s_{n-1}- s_{n-2}|[/itex]
 

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