The sequences
and
of complex numbers are the canonical numerators and denominators of some continued fraction
if and only if



for all . If so, then
is uniquely determined by:
i.e.
Example. Find a continued fraction with approximants 1/2, 3/4, 7/8, 15/16 etc.
for ,
.
So the sought continued fraction is
references
Lorentzen, Waadeland – Continued fractions with applications