Colloquium publications.

102

THE BOSTON COLLOQUIUM.

It might be thought that the result of Phragmen makes the concept of absolute summability useless. This is, however, in no
wise the case. At any rate, Borel employs the concept to establish the important conclusion that a divergent series, if absolutely
summable, can be manipulated precisely as a convergent series.
Thus if two absolutely summable series, whether convergent or
divergent, are multiplied together, the resultant series will also be
absolutely summable, and the function which it defines will be
the sum or product of the functions defined by the two former
series. Or, again, if an absolutely summable series is differentiated term by term, another such series is obtained, and the latter
yields a function which is the derivative of the one defined by
the former series. Lastly, the function determined by an absolutely summable series can not be identically zero, unless all the
coefficients of the series vanish.
These facts make possible the immediate application of Borel's
theory to differential equations. If, in short,
P(, y, y',..., y())is a differential equation which is holomorphic in x at the origin and
is algebraic in y and its derivatives, any absolutely summable
series (I), which satisfies formally the equation, defines an analytic
function that is a solution of the equation. For example, it will be
found that the series of Laguerre satisfies formally the equation
x27~( 1)y=i
x2d + () y _ 1,
and hence the function
~ e-'dz
f(x) = o - ze
must be a solution of the equation.
These conclusions of Borel should be strongly emphasized.
In any complete theory of divergent series it is an ultimatum
that they shall in all essential points * permit of manipulation
*In an absolutely summable series it is not always legitimate to change the
order of an infinite number of terms. Cf. Borel, Journ. de Math., ser. 5, vol. 2
(1896), p. 111.