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I am studying the following sequence, which comes from the secant method applied to $f : x \mapsto x^3$ between $\left[-\frac{1}{2},1\right]$ :

$$x_{n+1}=x_n - \frac{x_{n}-x_{n-1}}{x_n^3-x_{n-1}^3} x_n^3$$

with $x_0=-\frac{1}{2}$ and $x_1=1$.

I proved that $(x_n)_n$ is increasing and is always strictly negative. I also know that $(x_n)_n$ converges toward $0$.

I want to study the speed of convergence $\underset{n \rightarrow +\infty}{\lim} \frac{|x_{n+1}|}{|x_{n}|}$.

I have the following hint :pose $y_n=-\frac{1}{x_n}$ and study the sequences $(a_n)_n=(y_{n+1}^2-y_n^2)_n$. Then use Césaro's Lemma.

However i don't understant this hint because if I manage to show the convergence of $(a_n)_n$, i will be able to use Césaro's Lemma on $(a_n)_n$ and have a limit for $(y_n^2)_n$ by telescopic summation... which seems weird because $y_n^2=\frac{1}{x_n^2}$ should explose because $x_n \underset{n \rightarrow +\infty}{\longrightarrow} 0$

Any helps or hints are welcomed !

Update I understood how I'll be able to study the speed of convergence using Césaro's lemma. However I still need to find that $(a_n)_n$ is convergent.

Velobos
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1 Answers1

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I'm not so certain about the hints, but you can observe the ratio of consecutive terms, writing the secant method in symmetric form:

$$x_{n+1}=\frac{f(x_n)x_{n-1}-f(x_{n-1})x_n}{f(x_n)-f(x_{n-1})}=\frac{x_n^3x_{n-1}-x_{n-1}^3x_n}{x_n^3-x_{n-1}^3}$$

Let $t_n=x_{n+1}/x_n,~x_{n+1}=t_nx_n$ to get

$$t_n=\frac{t_{n-1}^2-1}{t_{n-1}^3-1}=\frac{t_{n-1}+1}{t_{n-1}^2+t_{n-1}+1}$$

which converges to $t\approx0.755$, the unique solution of

$$t^3+t^2-1=0$$

and hence

$$\lim_{n\to\infty}\frac{x_{n+1}}{x_n}=t\approx0.755$$

Simply Beautiful Art
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