What conditions must be met for the sequence defined by the Babylonian method to have a well-defined limit?

Conditions

• $a > 0$
• $x_0 > 0$

Reasoning, step by step

1. Examine the recurrence relation $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$.
2. Note that division by $x_n$ requires $x_n \neq 0$.
3. If $a > 0$ and $x_0 > 0$, then $x_1 > 0$, and by induction $x_n > 0$ for all $n$.
4. Verify that the solution strategy involves proving monotonicity and boundedness, which relies on these positivity constraints.
5. Conclude that under these conditions, the limit exists and equals $\sqrt{a}$.

Example

given $x_0 > 0$ and the recursive formula ... with constant $a > 0$, prove that the limit of the sequence exists as n approaches infinity, and find this limit.

Common misconceptions

• Thinking that $x_0$ can be negative (it would converge to $-\sqrt{a}$ if allowed, but the problem statement specifies $x_0 > 0$).
• Assuming $a=0$ leads to a meaningful square root calculation via this specific geometric iteration (it collapses trivially).

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Explore next

• What happens to the sequence if $x_0 < 0$?
• Is the condition $a > 0$ strictly necessary for the algebraic manipulation, or just for the geometric interpretation shown?