Real Sequence/Examples/Root (2 + Root x(n))
Jump to navigation
Jump to search
Example of Real Sequence
Let $\sequence {x_n}$ denote the real sequence defined as:
- $x_n = \begin {cases} \sqrt 2 : n = 1 \\ \sqrt {2 + \sqrt {x_{n - 1} } } & : n > 1 \end {cases}$
Then $\sequence {x_n}$ converges to a root of $x^4 - 4 x^2 - x + 4 = 0$ between $\sqrt 3$ and $2$.
Proof
Because $x_1 = \sqrt 2$, we have that:
- $x_2 > \sqrt 2 = x_1$
Suppose that:
- $x_n > x_n - 1$
for some $n \ge 2$.
Then:
\(\ds x_{n + 1}\) | \(=\) | \(\ds \sqrt {2 + \sqrt {x_n} }\) | ||||||||||||
\(\ds \) | \(>\) | \(\ds \sqrt {2 + \sqrt {x_{n - 1} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x_n\) |
Hence $\sequence {x_n}$ is seen to be strictly increasing.
Next we note that if $x_n > 1$:
\(\ds x_{n + 1}\) | \(=\) | \(\ds \sqrt {2 + \sqrt {x_n} }\) | ||||||||||||
\(\ds \) | \(>\) | \(\ds \sqrt {2 + \sqrt 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt 3\) |
As $x_n$ is strictly increasing, it follows that $x_n > \sqrt 3$ for all $n \ge 1$.
Similarly, we note that if $x_n < 2$:
\(\ds x_{n + 1}\) | \(=\) | \(\ds \sqrt {2 + \sqrt {x_n} }\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \sqrt {2 + \sqrt 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {2 + 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2\) |
So $\sequence {x_n} \to k$ where $\sqrt 3 < k \le 2$.
By definition of $x_n$:
\(\ds k\) | \(=\) | \(\ds \sqrt {2 + \sqrt k}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds k^2\) | \(=\) | \(\ds 2 + \sqrt k\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {k^2 - 2}^2\) | \(=\) | \(\ds x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds k^4 - 4 k^2 + 4\) | \(=\) | \(\ds k\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds k^4 - 4 k^2 - k + 4\) | \(=\) | \(\ds 0\) |
By investigating the shape of the graph, it is seen that this is the only root of $x^4 - 4 x^2 - x + 4 = 0$ strictly greater than $1$.
The other root, as is seen by inspection, is in fact $1$.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 14$