The object of this paper is to develop certain general theorems about convergence factors. In the case of series we shall mean by convergence factors a set of functions of a parameter which, when introduced as factors of the successive terms of the series, cause a divergent series to converge, or a series which is already convergent to converge more rapidly, throughout a certain range of values of the parameter. In the case of integrals we shall mean by a convergence factor a function of the variable of integration and a new parameter which, when introduced as a factor of the integrand, causes a divergent integral to converge, or an integral which is already convergent to converge more rapidly, throughout a certain range of valuies of the parameter. Although the name convergence factor is of recent origin, the subject itself, in the simple case of a convergent series, goes back to ABEL and virtually takes its rise in his well known theorem on the continuitv of a power series. The successive powers of x in the terms of a power series are, in fact, convergence factors of a simple nature, though not ordinarily regarded as such. The first attempt to extend the theory of convergence factors to general types of divergent series was made by FROBENIUS.t The class of series which he considered will be designated in this paper as summable seriest! and may be defined as follows: Given a series