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

by Various

p_n = a_np_n-1 + b_np_n-2, q_n = a_nq_n-1 + b_nq_n-2 ;

and the first two partial quotients are given by

b₁ = p₁, a₁ = q₁, b₁a₂ = p₂, a₁a₂ + b₂ = q₂.

If we form then the continued fraction in which p₁, p₂, p₃, ..., p_n are u₁, u₁ + u₂, u₁ + u₂ + u₃, ..., u₁ + u₂ + ..., u_n, and q₁, q₂, q₃, ..., q_n are all unity, we find the series u₁ + u₂ + ..., u_n equivalent to the continued fraction

u1		u2 ⁄ u1			u3 ⁄ u2			un ⁄ un-1
1	-	1 +	u2	-	1 +	u3	- ... -	1 +	un
			u1			u2			un-1

which we can transform into

u1		u2		u1u3		u2u4		un-2un	,
1	-	u1 + u2	-	u2 + u3	-	u3 + u4	- ... -	un-1 + un

a result given by Euler.

2. In this case the sum to n terms of the series is equal to the n^th convergent of the fraction. There is, however, a different way in which a Series may be represented by a continued fraction. We may require to represent the infinite convergent power series a₀ + a₁x + a₂x² + ... by an infinite continued fraction of the form

β0		β1x		β2x		β3x
1	-	1	-	1	-	1	- ...