Two Integral Sequences

We define the sequences $x_n=p_{3n}$ , $y_n=q_{3n}$ . Since the $3n$ -convergents will converge to the same real number that the $n$ -convergents do, $x_n/y_n$ also converges to the limit of the continued fraction. Each sequence $\{x_n\}$ , $\{y_n\}$ will obey the recurrence relation derived in the previous section (where $z_n$ is a stand-in for $x_n$ or $y_n$ ):

$$z_n=2(2n-1)z_{n-1}+z_{n-2} n\geq2.$$ (5.3.1)

The two sequences can be found in Table 5.1. (The initial conditions $x_0=1$ , $x_1=3$ , $y_0=y_1=1$ are taken straight from the first few convergents of the original continued fraction.) Notice that since we are skipping several convergents at each step, the ratio $x_n/y_n$ converges to $e$ very quickly.

Table 5.1: Convergents

$n$	0	1	2	3	4	$\cdots$
$x_n$	1	3	19	193	2721	$\cdots$
$y_n$	1	1	7	71	1001	$\cdots$
$x_n/y_n$	1	3	$2.714\ldots$	$2.71830\ldots$	$2.7182817\ldots$	$\cdots$