Evaluating $\sqrt{6+\sqrt{6+\cdots}}$

Question

Tough as introduction to analysis for beginners (Dutch handbook - I'm Belgian). Again ($n$) means index $n$, $x_1 = \sqrt6$, $x_{n+1} = \sqrt{6+x_n}$

1. Question:

$$|x_{n+1} - 3| \le 1/5 \cdot |x_n - 3|$$

For me this means that $3$ as a 'limit', we need to find that the distance between $x_{n+1}$ and the 'limit' is $1/5$ the distance between the $x_n$ and the limit. Where does the $1/5$ come from?

1. Prove that $|x_n - 3|\le (1/5)^{n-1}$

2. prove that the sequence converges to $3$.

ps: When I studied maths in 1980. we went quickly towards metric spaces, so these calculus minded times are nothing compared to those times. But still, as I didn't pass then, I'd like to restart on a new basis. Thanks for all the help. If you know where maths can be studied in community on the net, always welcome.

Answer 1

For the inequality, by the definition of $x_n$ we have
$$x_n-3=\sqrt{6+x_{n-1}}-3.$$ Multiply by $\dfrac{\sqrt{6+x_{n-1}}+3}{\sqrt{6+x_{n-1}}+3}$. So we are multiplying by $1$ in a fancy way. We get $$x_n-3=\frac{x_{n-1}-3}{\sqrt{6+x_{n-1}}+3}.\tag{$\ast$}$$ The bottom is clearly $>5$, since the $x_i$ start and stay positive. One can do better than $5$ here, for example we can without thought replace $5$ by $\sqrt{6}+3$, and with not much more by $\sqrt{6+\sqrt{6}}+3$. But it doesn't matter, $5$ is good enough for a proof of convergence. It would even be enough to observe that the denominator is $>3$. Taking absolute values, we find that $$|x_n-3|=\frac{|x_{n-1}-3|}{\sqrt{6+x_{n-1}}+3}<\frac{|x_{n-1}-3|}{5}.$$ Iterate. The distance to $3$ gets divided by at least $5$ with each iteration, so after a (short) while $x_n$ is awfully close to $3$. Thus our sequence has limit $3$.

Answer 2

We will also assume that $x_n$ lies between 0 and 3.

$3-x_{n+1} = 3 - \sqrt{6 + x_n} = 3 - \sqrt{9-(3-x_n}$.

Call $3-x_n$ as $a_n$.

So $a_{n+1} = 3 - \sqrt{9-a_n} = 3(1 - \sqrt{1-\frac{a_n}{9}}) \le 3\frac{a_n}{18} = \frac{a_n}{6}$ (by the Taylor series expansion).

Also, $a_{n+1}$ does not become negative like this, and as we will show its absolute value keeps decreasing. So we justified in assuming that $x_n$ lies between 0 and 3.

So $a_{n+1} \le \frac{a_n}{5}$ also

As $a_{n+1} \le \frac{a_n}{5}$, and $a_1$ = $3 - \sqrt{6} \lt 1$, $a_n \le \frac{1}{5}e^{n-1}$ in general.

So the series converges to 3.

Answer 3

Let's define the auxiliary sequence $a_{(n)}$, $n\ge1$, n in${\mathbb N}$ as follow:

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

i). Taking into account that $x_{n}$ is positive, one sees that $|\sqrt{6+x_n}+3|>5$. Hence, our first inequality may be proved as follows:

$$a_{n}=\frac{|\sqrt{6+x_n}-3|}{|x_n - 3|}=\frac{1}{|\sqrt{6+x_n}+3|}\le \frac{1}{5} \to \space a_{n}\le \frac{1}{5}. $$

ii). Proving the second inequality: $$a_{1}\cdot a_{2} \cdot a_{3}\cdots a_{n-1}=\frac{|x_{n} - 3|}{|\sqrt6 - 3|}\le \left({\frac{1}{5}}\right)^{n-1} \to \space |x_{n} - 3|\le {|\sqrt6 - 3|}\left({\frac{1}{5}}\right)^{n-1} \le \left({\frac{1}{5}}\right)^{n-1}.$$

iii). Using the inequality from the previous point we get immediately that:

$$\lim_{n\to\infty} |x_{n} - 3|\le 0 \to \lim_{n\to\infty} x_{n}=3.$$

The proof is complete.