Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 163

This norm will not be altered in any way by dividing (10)
through by SO. It is therefore determined uniquely by the ratios
of SO, SI, S2, and conversely the ratios by the norm.
Without loss of generality we may set So = 1. Place also
(11) S3 = + A+3 + Bn+3Sl + Q7+32,,
B C AA
P B n Q n i R n nB.
+ C + q+ A+ = A+ B+
nln+1 1n+l n+1 n+1
If then n + 3 in (11), is replaced successively by n and n + 1,
and the two equations are solved for S, and S2, we obtain
Q= q+ Cn+l S On S +1
^1- ------ pI
or
(12) S, (I = 0+ S. - C. So+),
n n
and
n n
(13) S- P,~ ( =n B + - B S).
An examination of Pa, Qn, RIt, X, A, will show that their degrees
in x are
n- 1, n -2, n-3 ---, - r. (n = 2r),
n- 1, n- 2, n-3, -r-1, -r- 1 (n= 2r+l ).
Hence the expansions of QJJP,, and RIjPn in descending powers of
x, agree with S, and S2 to terms of degree 3r - 1 and 3r - 2 inclusive if n= 2r, and of the 3rth degree if n-=2r + 1. The
generalized continued fraction therefore affords a solution of the
problem: to find two rational fractions with a common denominator which shall give as close an approximation to the given
functions S, and S2 as is consistent with the degrees prescribed for
their numerators and denominators.
When three series in ascending powers of x,

S, = k?() + kx',x + k() +. 1..

(i = 1, 2, 3),