Question

Qno. 3) I) let X1=8 and Xn+1=1/2Xn+2 for all n belongs to N .show that X is bounded and monotone.also find the limit.
ii) let X1>1 and Xn+1=2-1/Xn for all n belongs to N. Show that (Xn) is bounded and monotone . Find the limits.

Answer 1

Answer on Question #82487 – Math – Real Analysis

Question

1) Let $x_1 = 8$ and $x_{n+1} = 2 + x_n / 2$ for all $n$ belongs to $N$. Show that $X$ is bounded and monotone. Also find the limit.

Solution

1) If $x > 4$ then $2 + \frac{x}{2} > 4$, so $x_n > 4$ (bounded from below by 4)

If $x_n > 4 \Rightarrow \frac{x_n}{2} > 2 \Rightarrow x_n - \frac{x_n}{2} > 2 \Rightarrow x_n > 2 + \frac{x_n}{2} \Rightarrow x_n > x_{n+1}$

So $x_n$ is monotone decreasing and bounded by 4.

Then it has some limit, say: $a = \lim_{n\to \infty}x_n$

Then $a = \lim_{n\to \infty}x_n = \lim_{n\to \infty}x_{n + 1} = 2 + \frac{a}{2}$, so $a = 2 + a / 2$.

So $a = 4$.

Question

2) Let $x_1 > 1$ and $x_{n+1} = 2 - 1/x_n$ for all $n$ belongs to $N$. Show that $\{x_n\}$ is bounded and monotone. Find the limits.

Solution

If $x > 1 \Rightarrow \frac{1}{x} < 1 \Rightarrow -\frac{1}{x} > -1 \Rightarrow 2 - \frac{1}{x} > 1$

So if $x_1 > 1$ then every $x_n > 1$

Let's prove, that for every $x > 0 \Rightarrow x > 2 - 1/x$

$(x - 1)^2 > 0$, so $x^2 > 2x - 1$, so $x > 2 - 1/x$

(we could divide both parts by $x$, because $x > 0$)

so $x_{n+1} < 2 - 1/x_n$. So $x_n$ is monotone decreasing and bounded by 1 from below.

Then it has some limit, say: $a = \lim_{n\to \infty}x_n$

Then $a = \lim_{n\to \infty}x_n = \lim_{n\to \infty}x_{n + 1} = 2 - \frac{1}{a}$

So $(a = 2 - 1/a)$, so $a = 1$.

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