Infinite Continued Fractions

This section begins with the continued fraction procedure, which associates to a real number $x$ a sequence $a_0,a_1,\ldots$ of integers. After giving several examples, we prove that $x = \lim_{n\rightarrow \infty} [a_0,a_1,\ldots,a_n]$ by proving that the odd and even partial convergents become arbitrarily close to each other. We also show that if $a_0,a_1,\ldots$ is any infinite sequence of positive integers, then the sequence of $c_n=[a_0,a_1,\ldots,a_n]$ converges, and, more generally, if $a_n$ is an arbitrary sequence of positive reals such that $\sum_{n=0}^{\infty} a_n$ diverges then $(c_n)$ converges.