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Let $c>0,a_1=c/2,a_{n+1}=\frac{1}{2}(c+a_n^2)$. Determine $c$ for which the sequence converges, for each $c$, and find $\lim_{n\to\infty} a_n$.

I have the feeling that I should use induction but I don't know how to start.

user91011
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Diane Vanderwaif
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1 Answers1

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You can use the fixed point theorem here. This states that if an iteration

$a_{n+1} = f(a_{n})$

has a stationary point $a_{*}$ such that $a_{*} = f(a_{*})$ and $|f'(a_{*})| < 1$ holds then $a_{*}$ is a point of attraction.

To start with you need to find $a_{*}$ such that $a_{*} = \displaystyle \frac{1}{2}(c + a_{*}^2)$

and then find which values of $c$ allow for $|f'(a_{*})| < 1$

(Note that there may be several fixed points of the iteration)

Keeran Brabazon
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  • we haven't learnt about fix point or iteration yet. – Diane Vanderwaif Sep 19 '13 at 23:03
  • $a_{n+1}=\frac{1}{2}(c+a_n^2)$ defines an iteration for $a_{n+1}$. A point of convergence of the sequence $a_{1},a_{2},\ldots$ must have the property that it is a stationary point, i.e. that $a_{} = 1/2(c+a_{}^{2})$ otherwise the sequence is divergent. I hope that makes things more clear – Keeran Brabazon Sep 19 '13 at 23:11