Colloquium publications.

80

THE BOSTON COLLOQUIUM.

f(x) = 0 we have the functions e-/x, e-1/2, * *, which fulfill the
assigned conditions. They are, namely, unlimitedly differentiable
within a positive interval terminating in the origin, and when x
approaches the origin from within this interval, the functions and
their derivatives have the limit 0. From this it follows immediately that if values other than zero be prescribed for the an, the
function will not be uniquely determined, since to any one determination we may add constant multiples of e1/x, e-/x2, **.
Inasmuch as the correspondence between the function and the
series is not reversibly unique, the series can not be used, in
general, for the computation of the value of the generating function. Nevertheless, although this is the case, the series is not
without its value. For consider the first m terms, mn being a
fixed integer. If x is sufficiently diminished, in value, each of
these terms can be made as small as we choose in comparison with
the one which precedes it, and the series therefore at the beginning has the appearance of being rapidly convergent, even though
it be really divergent. Evidently also as x is decreased, it has
this appearance for a greater and greater number of terms, if not
throughout its entire extent. Now by hypothesis the generating
function was unlimitedly differentiable within the interval, and
the successive derivatives are consequently continuous within (0, a).
Hence if the interval is sufficiently contracted, f(-+l)(x)/(m + 1)!
can be made as nearly equal to a+nt1 throughout the interval as is
desired. We have then for the remainder in Taylor's formula:
(4) R,+1(x) = (m 1 a+ 'l( + <) ( e),
in which e is an arbitrarily small positive quantity. Consequently
if the first m + 1 terms of the series should be used to compute
the value of the generating function, the error committed would
be approximately equal to the next term, provided x be taken sufficiently small.
In these considerations there is, of course, nothing to indicate
when x is sufficiently small for the purpose. If the result holds