Continued Fractions, Convergence Theory. Vol. 1, 2nd Editions. Loretzen, Waadeland. Atlantis Press. 2008

6 Chapter 1: Introductory examples
the word “ approximant” always meant
f n := S n (0) (1.2.3)
whereas S n (w n ) was referred to as a “ modified approximant” where w n was the
“ modifying factor”. Classical approximants f n = S n (0) play a special role in the
theory. In particular, the concept of convergence is based on {f n }:
✗
Definition 1.2. A continued fraction converges (in the classical sense) to
a value f ∈ Ĉ if lim f n = f.
✖
✔
✕
If K(a n /b n ) fails to converge, we say that it diverges.
A classical approximant f n is obtained by truncating the continued fraction after n
fraction terms. The part we cut away,
f (n) := a n+1
b n+1 +
a n+2
b n+2 +
a n+3
b n+3 + ···
(1.2.4)
is called the nth tail of b 0 + K(a n /b n ). This is also a continued fraction, and it
converges if and only if b 0 + K(a n /b n ) converges. Indeed, (1.2.4) converges to f (n)
if and only if b 0 + K(a n /b n ) converges to
f = S n (f (n) ). (1.2.5)
The sequence {f (n) } is then called the sequence of tail values for b 0 + K(a n /b n ).
This sequence will be important in our investigations.
✬
✩
Lemma 1.1. Let S n be given by (1.2.1). Then
where
S n (w) = A n−1w + A n
B n−1 w + B n
for n =1, 2, 3,... (1.2.6)
A n = b n A n−1 + a n A n−2 , B n = b n B n−1 + a n B n−2 (1.2.7)
with initial values A −1 =1, A 0 = b 0 , B −1 =0and B 0 =1.
✫
✪
Proof :
It is clear that
S 0 (w) =b 0 + w = b 0 + w
1+0w , S 1(w) =b 0 + a 1
b 1 + w = b 0b 1 + a 1 + b 0 w
,
b 1 + w