# Convergent Real Sequence/Examples/x n = root x n-1 y n-1, 1 over y n = half (1 over x n + 1 over y n-1)

Jump to navigation
Jump to search

## Example of Convergent Real Sequence

Let $\sequence {x_n}$ and $\sequence {y_n}$ be the real sequences defined as:

- $x_n = \begin {cases} \dfrac 1 2 & : n = 1 \\ \sqrt {x_{n - 1} y_{n - 1} } & : n > 1 \end {cases}$

- $\dfrac 1 {y_n} = \begin {cases} 1 & : n = 1 \\ \dfrac 1 2 \paren {\dfrac 1 {x_n} + \dfrac 1 {y_{n - 1} } } & : n > 1 \end {cases}$

Then both $\sequence {x_n}$ and $\sequence {y_n}$ converge to the limit $\dfrac \pi 4$.

## Proof

By definition, we have that:

- $\sqrt {x_{n - 1} y_{n - 1} }$ is the geometric mean of $x_{n - 1}$ and $y_{n - 1}$

- $\dfrac 1 2 \paren {\dfrac 1 {x_n} + \dfrac 1 {y_{n - 1} } }$ is the reciprocal of the harmonic mean of $x_n$ and $y_{n - 1}$.

We are given that:

- $\dfrac 1 2 = x_1 < y_1 = 1$

Assuming $x_{n - 1} < y_{n - 1}$, it follows from Geometric Mean of two Positive Real Numbers is Between them that:

- $x_{n - 1} < x_n < y_{n - 1}$

Given that $x_n < y_{n - 1}$, it follows from Harmonic Mean of two Real Numbers is Between them that:

- $x_n < y_n < y_{n - 1}$

It follows from the Principle of Mathematical Induction that:

- $x_{n - 1} < x_n < y_n < y_{n - 1}$ for $n = 2, 3, \ldots$}

Hence we have:

- $\sequence {x_n}$ is strictly increasing and bounded above by $y_1 = 1$.

- $\sequence {y_n}$ is strictly decreasing and bounded below by $x_1 = \dfrac 1 2$.

So by the Monotone Convergence Theorem (Real Analysis), both $\sequence {x_n}$ and $\sequence {y_n}$ converge.

Let:

- $x_n \to l$ as $n \to \infty$
- $y_n \to m$ as $n \to \infty$

Then:

- $l^2 = l m$

and:

- $\dfrac 1 m = \dfrac 1 2 \paren {\dfrac 1 l + \dfrac 1 m}$

from which it follows that:

- $l = m$

It remains to be shown that $l = m = \dfrac \pi 4$.

This needs considerable tedious hard slog to complete it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 5$: Subsequences: Exercise $\S 5.7 \ (4)$