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

30 Chapter 1: Introductory examples
1.4 Correspondence between power series and continued
fractions
1.4.1 From power series to continued fractions
A continued fraction of the form
b 0 + a 1z a 2 z a n z
1 + 1 +···+ 1 +··· ; 0 ≠ a n ∈ C (4.1.1)
is called a regular C-fraction. (This concept has no particular connection to regular
continued fractions.) Regular C-fractions often work better than power series as
far as speed of convergence and domain of convergence are concerned. Hence it is
of interest to go from a power series to a continued fraction of this particular form.
One way of doing this was demonstrated in the previous section. A more primitive
way is the method of successive substitutions ([Lamb61]) due to Lambert (1728 -
1777), a colleague of Euler and Lagrange in Berlin. We illustrate this method by
an example:
Example 13. We shall compute the circumference L of the ellipse
x 2
a + y2
=1, a ≥ b ≥ 0 , a > 0 . (4.1.2)
2 b2 The well known arc length formula leads to the elliptic integral
∫ π/2 √
L =4a 1 − ε 2 sin 2 θdθ
0
where ε := √ a 2 − b 2 /a is the eccentricity of the ellipse, and thus b 2 = a 2 (1 − ε 2 ).
By setting
( a − b
) 2
t := , (4.1.3)
a + b
and expanding the integrand in a series and integrate, we get
L = π(a + b)
∞∑
( ) 2 1/2
t n = π(a + b)(
1+ t
n
2 + t2
2 2 + t3
6 2 + 25t4 + ···)
, (4.1.4)
8 214 n=0
([Hütte55], [LoWa85]). One way of finding approximate values for L is to truncate
the series. But we can also transform the series into a continued fraction:
1+ t
2 2 + t2
2 6 + t3 25 t4
+
28 2 14 + ···=1+ t/2 2
(
1+ t
2
+ t2
4 2
+ 6
25 t3
2 12 + ···
) −1