Problem #PRU-61324

Problems Calculus Number sequences Limit of a sequence, convergence Algebra and arithmetic Mean values Sequences Recurrent relations Recurrent relations (other)

15–16 3.0

Problem

We call the geometric-harmonic mean of numbers $a$ and $b$ the general limit of the sequences ${a_{n}}$ and ${b_{n}}$ constructed according to the rule $a_{0} = a$ , $b_{0} = b$ , $a_{n + 1} = \frac{2 a_{n} b_{n}}{a_{n} + b_{n}}$ , $b_{n + 1} = \sqrt{a_{n} b_{n}}$ ( $n \geq 0$ ).

We denote it by $ν (a, b)$ . Prove that $ν (a, b)$ is related to $μ (a, b)$ (see problem number 61322) by $ν (a, b) \times μ (1 / a, 1 / b) = 1$ .

Problem number 61322 says that both of these sequences have the same limit.

This limit is called the arithmetic-geometric mean of the numbers $a, b$ and is denoted by $μ (a, b)$ .

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