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Convergence of a positive simple continued fraction

When

b_{j} = 1

and

a_{j}

are positive integers, adjacent convergents satisfy

p_{n} q_{n - 1} - q_{n} p_{n - 1} = (- 1)^{n + 1} .

Even and odd convergents are monotone from opposite sides, their separation is

1/ (q_{n} q_{n + 1})

, and

q_{n} \to \infty

. They therefore have a common limit

x

, with

\frac{p _{n}}{q _{n}} - x + \frac{p _{n + 1}}{q _{n + 1}} - x = \frac{1}{q _{n} q _{n + 1}} .

Table of contents 72 1
- All-one continued fraction and Fibonacci ratios Convergence of a positive simple continued fraction 32

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