Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 97

The series is said by Borel to be summable * at a point x when the
integral (4) has a meaning at this point.
The second case is that in which the fundamental series is
divergent. The associated series in this case may be either convergent or divergent. If it is convergent only over a portion of
the plane of u = zx, we are to understand by F(u) not merely the
value of the associated series but of its analytic continuation.
Let x for an instant be given a fixed value. Then when z
describes the positive real axis, u in its plane describes the ray
from the origin passing through the point x. If F(u) is holomorphic along this ray, it is possible that the integral (4) will have a
sense. Suppose that this holds good as long as x lies within a
certain specified region of its plane. Then for this region a function will be obtained uniquely from the divergent series by the use
of the integral, precisely as in the case of the series of Laguerre.
This method of treatment is obviously restricted to divergent
series for which -the associated series are convergent, and it will
not always be applicable even to these. A divergent series in which
there is an infinite number of coefficients of the same order of magnitude as the corresponding coefficients of
(6) 1 + x + (2!)22 + (3!)2x3 +... + (n!) +...
can not be summed in this manner. It will be noticed, however,
that the series just given is one whose first associated series is the
series of Laguerre, and whose second associated series is consequently convergent.
The method of Borel can be readily extended so as to take
account also of such series, or, more generally, of series that have
an associated series of the nth order which is convergent. One
mode of doing this is by the introduction of an n-fold integral.
Suppose, for example, that in (6) one of the two factorials n! is
replaced by
X e-zzndz
* Some other term would be preferable since his definition refers only to one
of many possible modes of summation. A series may be simultaneously " summable" at a point x by one method, and non-summable by another.