Summation of Series. | 407 
différent methods. Mr. Nicoias Bernovixr and Mr. Mom- 
MoRT invefligated the fum of the feries (P) A+Br4Cr+ 
ih Ug? " 
@e. Hy a feries 4 ): — et ep etek s 
where d’, d”, d’”’, &c. denote the fucceflive differences of 
the terms A, B, C, D, &c. If r be negative, the denomi- 
nators become +7, (1+7), (1+7)’, &c. 
It has been. obferved, in the Meditationes, that in fwift con- 
verging feries. the feries P will converge more fwiftly than the 
feries Q ; in feries converging according to a geometrical ratio, 
fometimes the one will converge more {wift, and fometimes 
the other. Jn other feries, which converge more flow, where 
moft commonly 7 nearly=1, it cannot in general be faid, 
which of the fericfes will converge the {wifteft. The preceding 
remark, viz. the addition of the firftterms of the feries, is ne- 
ceflary in moft cafes. of finding the fums by {feriefes of this. 
kind. 
It is not unworthy of obfervation, that in almoft all cafes of 
infinite feries, the convergency depends on the roots of the 
given equations, which remark was firft publifhed in the Me- 
ditationes. For example: in finding approximates to the roots 
of given equations the convergency depends on how much the: 

approximates given are more near to one root than to any 
other ; and confequently, when two or more roots or values of ~ 
an unknown quantity are nearly equal, different rules are to be 
applied, which are improvements of the rule of falfe.. This. 
rule, and the above-mentioned obfervations were firft given in the 
Meditationes Algebraic et Analytica, with. feveral other: 
additions on fimilar fubjects.. 
Many 
