28 “ 
14+24+44...=-1, 
while by others they are regarded as meaningless results, and 
have thus been altogether rejected from analysis. 
It is impossible to avoid the occurrence of these series : 
they present themselves at a very early stage of algebra, in 
the form of geometrical progressions and binomial develop- 
ments; and thenceforward are continually met with by the 
analyst up to the remotest applications of the integral calculus. 
The existing vagueness and indecision, as to the proper mode 
of interpreting such series, is thus a matter of some concern, 
as calculated to retard the progress of science, to diminish our 
confidence in some of the truths of analysis, and to give cur- 
rency to results involving error and contradiction. 
In the present communication it will be my endeavour to 
ascertain the causes of the perplexities and discrepancies above 
adverted to, and to discover the legitimate interpretation of 
diverging infinite series; from which it will, I think, follow | 
that certain expressions received into analysis as the sums of 
several of these, are erroneous. The fact that Poisson, Cauchy, 
Abel, and indeed most of the modern continental writers, re- 
ject diverging infinite series, and pronounce them to have no 
sums, does not render such an endeavour the less necessary ; 
inasmuch as the analytical operations, in virtue of which finite 
values have been attributed to extensive classes of these series 
by Euler and subsequent investigators, remain, I believe, un- 
impugned. Widely different methods appear to concur in 
furnishing the same numerical results for such series; as, for 
instance, the method of definite integrals, and that deduced 
from the differential theorem, both so frequently applied by 
Euler to effect the summations of series of this kind; and the 
numerical results obtained by him have often, apparently, been 
verified by later computers; some of whom have employed 
methods quite distinct from those of Euler; as, for instance, 
