Encyclopaedia Britannica, 11th Edition, "Constantine Pavlovich" to "Convention" / Volume 7, Slice 2

by Various

Let un ≡	pnr+m	; then un =	a1		a2		ar	,
	qnr+m		b1	+	b2	+ ... +	br + un1

leading to an equation of the form Au_nu_n-1 + Bu_n + Cu_n-1 + D = 0, where A, B, C, D are independent of n, which is readily solved.

2. The Convergence of Infinite Continued Fractions.—We have seen that the simple infinite continued fraction converges. The infinite general continued fraction of the first class cannot diverge for its value lies between that of its first two convergents. It may, however, oscillate. We have the relation p_nq_n-1 - p_n-1q_n = (-1)ⁿb₂b₃...b_n, from which

pn	-	pn-1	= (-1)n	b2b3 ... bn	,
qn		qn-1		qnqn-1

and the limit of the right-hand side is not necessarily zero.

The tests for convergency are as follows:

Let the continued fraction of the first class be reduced to the form

d1 +	1		1		1
	d2	+	d3	+	d4	+ ...,

then it is convergent if at least one of the series d₃ + d₅ + d₇ + ..., d₂ + d₄ + d₆ + ... diverges, and oscillates if both these series converge.

For the convergence of the continued fraction of the second class there is no complete criterion. The following theorem covers a large number of important cases.