Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 113

This function has no singular points other than those which are
possible for the m series from which it was derived. When r
different series
a0 + aix + a2x2 + *,
b, + bx + b2x2 +...,
ro + rIx + r2x2 +...,
are used, a similar conclusion is reached for the series
G(a0, b0,. *, t0) + G(a,, b..., r,)x + G(a2, b,..., r)x2 +...
where G denotes a polynomial in which the constant term is lacking.
These results are of particular interest when applied to the
series
(10) 1+ x + 2x2.+.. "x...
and
2 9n
(11) I + x + +.. - +...
which are the expansions of 1 + x/(1 - x)2 and log (1 + x). Since
these functions have only one singular point, x = 1, in the finite
plane, the only possible singularities of
(are (n-
are x= 0, 1, oO.*
The continued repetition of the above process for combining
series leads naturally to a consideration of series of the form
(12) YP(a)x"
in which a convergent power series P(u) appears in place of the
polynomial G(u). Various theorems concerning cases of this
* Obviously a constant term can be included now in the polynomial G(n, 1/n).