Pick two positive numbers \(a_{0}\) and \(b_{0}\) with \(a_{0}>b_{0}\) and write out
the first few terms of the two sequences \(\left\\{a_{n}\right\\}\) and
\(\left\\{b_{n}\right\\}:\)
$$a_{n+1}=\frac{a_{n}+b_{n}}{2}, \quad b_{n+1}=\sqrt{a_{n} b_{n}}, \quad \text
{ for } n=0,1,2 \dots$$
(Recall that the arithmetic mean \(A=(p+q) / 2\) and the geometric mean
\(G=\sqrt{p q}\) of two positive numbers \(p\) and \(q\) satisfy \(A \geq G\).
a. Show that \(a_{n}>b_{n}\) for all \(n\).
b. Show that \(\left\\{a_{n}\right\\}\) is a decreasing sequence and
\(\left\\{b_{n}\right\\}\) is an increasing sequence.
c. Conclude that \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\)
converge.
d. Show that \(a_{n+1}-b_{n+1}<\left(a_{n}-b_{n}\right) / 2\) and conclude that
\(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n} .\) The
common value of these limits is
called the arithmetic-geometric mean of \(a_{0}\) and \(b_{0},\) denoted
\(\mathrm{AGM}\left(a_{0}, b_{0}\right)\).
e. Estimate AGM(12,20). Estimate Gauss' constant \(1 / \mathrm{AGM}(1,
\sqrt{2})\).
