Arithmetic-geometric mean

Prove that the sequences $x_{n}$ and $y_{n} (n=1,2,\ldots )$ determined by the following formulas

have a common limit

(the arithmetic-geometric mean of the numbers $a$ and $b$).

Numbers $a$ and $b$ are given, with $a > b > 0$. New pairs of numbers are then formed by the formulas: $a_{n+1} = \frac{a_{n}+b_{n}}{2}$, $b_{n+1} = \frac{2 a_{n} b_{n}}{a_{n}+b_{n}}$. Here $a_{0} = a$ and $b_{0} = b$. Prove that $a_{n} b_{n} = ab$, that

and, finally, that $\lim_{n \to \infty } a_{n} = \lim_{n \to \infty } b_{n} = \sqrt{ab}$.

From two given positive numbers $a$ and $b$ one can construct a series of new numbers using the equalities: $a_{0} = a$, $b_{0} = b$, $a_{1} = \frac{a_{0}+b_{0}}{2}$, $b_{1} = \sqrt{a_{0} b_{0}}$, and in general, $a_{n+1} = \frac{a_{n}+b_{n}}{2}$, $b_{n+1} = \sqrt{a_{n} b_{n}}$. Prove that as $n \to \infty$ the numbers $a_{n}$ and $b_{n}$ tend to a common limit. (Gauss's arithmetic-geometric mean.)

Two sequences of numbers are formed by the formulas:

Here $a_{0} = a$, $b_{0} = b$ are given positive numbers. Find $\lim_{n \to \infty } a_{n}$, which equals $\lim_{n \to \infty } b_{n}$, setting $a = b \cos \varphi$ for $a < b$ and $a = b \operatorname {ch} \varphi$ for $a > b$.

The sequences $\left\{ x_{n}\right\}$ and $\left\{ y_{n}\right\}$ are given recursively:

Prove that:

Prove that if

then:

The sequences $\left\{ x_{n}\right\}$ and $\left\{ y_{n}\right\}$ satisfy the conditions:

Prove Gauss's formula

Let $a > b > 0, a_{1} = (a+b) / 2, b_{1} = \sqrt{a b}, a_{n} = \left(a_{n-1}+b_{n-1}\right) / 2, b_{n} = \sqrt{a_{n-1} b_{n-1}}, n = 2,3, \ldots$ Then there exists a finite limit (see problem 244,1) in $[1, \S 8]$)

Prove that $\int_{0}^{\pi / 2} \frac{d \varphi }{\sqrt{a^{2} \cos^{2} \varphi +b^{2} \sin^{2} \varphi }} = \frac{\pi }{2 c(a, b)}$.