Arithmetic–Geometric Mean Iteration: Monotonicity and Convergence

A classic way konkur builds a convergence problem is to interleave two sequences through the geometric mean and the arithmetic mean. Starting from $0 < a_1 < b_1$, define

$a_{n+1} = \sqrt{a_n b_n}, \qquad b_{n+1} = \frac{a_n + b_n}{2}.$

The whole behaviour is driven by one inequality: the AM–GM inequality, $\sqrt{xy} \le \tfrac{x+y}{2}$ for $x, y \ge 0$, with equality iff $x = y$.

The pattern to recognize

Whenever a problem mixes $\sqrt{a_n b_n}$ (or any geometric mean) with $\tfrac{a_n+b_n}{2}$ (or any arithmetic mean), you are looking at the arithmetic–geometric mean (AGM) iteration. It is a nested-interval machine: $[a_n, b_n]$ is a shrinking chain of intervals.

How to apply it (4 moves)

1. Order the terms. AM–GM gives $a_{n+1} = \sqrt{a_n b_n} \le \tfrac{a_n + b_n}{2} = b_{n+1}$, so $a_n \le b_n$ for every $n$ (the hypothesis $a_1 < b_1$ seeds it).
2. Show $\{a_n\}$ increases. Because $b_n \ge a_n$, $\;a_{n+1} = \sqrt{a_n b_n} \ge \sqrt{a_n \cdot a_n} = a_n.$
3. Show $\{b_n\}$ decreases. Because $a_n \le b_n$, $\;b_{n+1} = \tfrac{a_n + b_n}{2} \le \tfrac{b_n + b_n}{2} = b_n.$
4. Invoke the Monotone Convergence Theorem. $\{a_n\}$ is increasing and bounded above by $b_1$; $\{b_n\}$ is decreasing and bounded below by $a_1$. Both converge.

Why the limits are equal

The gap collapses. Subtracting,

$b_{n+1} - a_{n+1} \le b_{n+1} - a_n = \frac{a_n + b_n}{2} - a_n = \frac{b_n - a_n}{2},$

so the width at least halves each step: $b_n - a_n \le \tfrac{b_1 - a_1}{2^{\,n-1}} \to 0$. Hence $\lim a_n = \lim b_n$. This common value is the arithmetic–geometric mean $M(a_1, b_1)$, squeezed between the two sequences forever:

$a_1 < a_2 < \cdots < M(a_1,b_1) < \cdots < b_2 < b_1.$

Mini-example

Take $a_1 = 1,\ b_1 = 3$.

The $a$-column rises, the $b$-column falls, and after only two steps they agree to two decimals — both heading to $M(1,3) \approx 1.863$.

Exam trick: the single fact $\mathrm{GM} \le \mathrm{AM}$ pins down both monotonicities at once — geometric-mean term goes up, arithmetic-mean term goes down. Don't test numerically; quote AM–GM plus the Monotone Convergence Theorem and you are done.