Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 137

q=0
q= 1
q=2
q= 3

p=O p-= p 2
oo~= -Go C t _ C 0+ GX + C I+S2,,,
- 00 1020.
-0oo NI N21 |
DoliDl D21
N02 1X12 N22
D02 D12 22

When the order of approximation of a rational fraction, taken
in its lowest terms, is exactly equal to the sum of the degrees of
numerator and denominator, increased by 1, the fraction will be
found once and only once in the table. If, conversely, a fraction
Npq/Dpq occurs but once in the table, the numerator and denominator are of degree p and q respectively, and the order of the
approximation which the fraction affords is exactly p + q + 1.
The approximant is then said by Pad4 to be normal. We shall
also call the table normal when it consists only of normal fractions,
or, in other words, when no approximant occurs more than once in
the table.
Obviously all approximants which lie upon a line perpendicular
to the principal diagonal of the table correspond to the same
value of p + q + 1. Hence in a normal table they approximate
to (4) in equal degree, and accordingly may be said to be equally
advanced in the table. If p + q + 1 increases in passing from
one fraction to another, the latter is the more advanced.
Two approximants will be called contiguous if the squares of
the table in which they are contained have either an edge or a
vertex in common.
Consider now a normal table, and take any succession of approximants, beginning with one upon the border of the table and passing always from one approximant to another which is contiguous
to it but more advanced. Pade shows that any such sequence of
approximants makes a continued fraction of which the approxi