What is the mathematical justification for the convergence of the Babylonian method sequence?

The mathematical justification relies on the Monotone Convergence Theorem. The sequence

x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})

is shown to be bounded below by

\sqrt{a}

(using the AM-GM inequality) and monotone decreasing for

n \ge 1

(assuming

x_1 > \sqrt{a}

). A bounded monotone sequence must converge to a limit

L

. By taking the limit on both sides of the recurrence relation, we get

L = \frac{1}{2}(L + \frac{a}{L})

, which solves to

L^2 = a

, hence

L = \sqrt{a}

(since

L>0

). This proves both the existence of the limit and its value.

Conditions

The constant $a > 0$ .
The initial guess $x_0 > 0$ .
The sequence is defined by $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$ .

Reasoning, step by step

Apply the AM-GM inequality to show $x_{n+1} \ge \sqrt{a}$ for all $n \ge 0$ .
Show that if $x_n \ge \sqrt{a}$ , then $x_{n+1} \le x_n$ , establishing monotonicity for $n \ge 1$ .
Invoke the Monotone Convergence Theorem to conclude that $\lim_{n \to \infty} x_n$ exists. Let this limit be $L$ .
Take the limit of the recurrence relation: $L = \frac{1}{2}(L + \frac{a}{L})$ .
Solve for $L$ : $2L = L + \frac{a}{L} \implies L = \frac{a}{L} \implies L^2 = a$ .
Since $x_n > 0$ , $L = \sqrt{a}$ .
Conclude that the sequence converges to $\sqrt{a}$ .

Example

The video states: 'The solution strategy involves proving monotonicity and boundedness.' It further notes that 'Since the sequence is bounded below by $\sqrt{a}$ (for $n \ge 1$ ) and decreasing (after the first step), it must converge. Solving the limit equation confirms the value is exactly the square root.'

Common misconceptions

Believing that boundedness alone is sufficient for convergence.
Thinking that the limit could be $-\sqrt{a}$ .
Confusing the geometric intuition with the rigorous analytical proof.

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What is the mathematical justification for the convergence of the Babylonian method sequence?

Conditions

Reasoning, step by step

Example

Common misconceptions

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What is the mathematical justification for the convergence of the Babylonian method sequence?

Conditions

Reasoning, step by step

Example

Common misconceptions

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