# Finding the limit of a sequence.

1. Aug 29, 2012

### ashwinnarayan

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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 $s_1$ and $s_2$ but the sequence is neither increasing nor decreasing.

For proving the convergence I tried used the Cauchy convergence test.
$|\frac{1}{2}(s_n + s_{n-1}) - s_n| = |\frac{1}{2}(s_{n-1} - s_n)| < \in$ 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 $s_n = f(s_n, s_{n-1})$.

2. Aug 29, 2012

### HallsofIvy

Staff Emeritus
$(s_n+ s_{n-1})/2$ is exactly half way between $s_n$ and $s_{n-1}$ That means that the distance between two consecutive terms is half the distance between the preceding termS: $|s_n- s_{n-1}|= (1/2)|s_{n-1}- s_{n-2}|$