Proving a sequence defined by a recurrence relation converges

sequences-and-series convergence-divergence recurrence-relations

How can I prove that this iterative sequence converges to $2$? Can I use the convergence definition?

$$a_{n+1} = \frac{a_n}{2} + \frac{2}{a_n},\qquad a_0 = 4 $$

Thanks for the help.


Solution 1:

Overall outline:

  • Show that $(a_n)_n$ is monotone decreasing, and positive, and hence convergent.
  • Show that the only limit possible is positive and satisfies $\ell = \frac{\ell}{2}+\frac{2}{\ell}$
  • Conclude.

Detailed outline: let $f\colon (0,\infty)\to (0,\infty)$ be the function defined by $f(x) = \frac{2}{x}+\frac{x}{2}$.

  • Observe that $a_n > 0$ for all $n\geq 0$ (e.g., by induction, but it's immediate).
  • Show that $f$ has a unique minimum on $(0,\infty)$, and that this minimum is $2$.
  • This implies $a_{n+1} = f(a_n) \geq 2$ for all $n\geq 0$.
  • Observe that $a_{n+1}-a_n = \frac{2}{a_n} - \frac{a_n}{2} \leq 0$ (e.g., by studying the function $g(x) = \frac{2}{x}-\frac{x}{2}$ on $[2,\infty)$). This gives you the monotonicity of $(a_n)_n$.
  • Now that you know that the sequence converges to some $\ell \geq 2$, by continuity of $f$ this limit must satisfy $\ell = f(\ell)$. Solve the equation, and conclude.

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