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

by Various

u_n+1 = 2u_n + (2n-1)²u_n-1,

whence

u_n+1 - (2n + 1)u_n = -(2n - 1){u_n - (2n - 1)u_n-1},

and we readily find that

pn	= 1 -	1	+	1	-	1	+ ... ±	1	,
qn		3		5		7		2n + 1

whence the value of the fraction taken to infinity is ¼π.

It is always possible to find the value of the n^th convergent to a recurring continued fraction. If r be the number of quotients in the recurring cycle, we can by writing down the relations connecting the successive p’s and q’s obtain a linear relation connecting

p_nr+m, p_(n-1)r+m, p_(n-2)r+m

in which the coefficients are all constants. Or we may proceed as follows. (We need not consider a fraction with a non-recurring part). Let the fraction be

a1		a2		ar		a1
b1	+	b2	+ ... +	br	+	b1	+ ...