172 Mr-G ompertz on Series which may be summed 
the terms, omitting 2, 3, & c. terms ; in fact, if r be a whole 
number, and the terms of the series be all positive, or any 
how positive and negative by sets, provided the same signs 
return in the same order after every set, consisting of r 
number of terms ; by continually multiplying by 2 sine of 
we shall get new series by taking the differences of the 
coefficients of every term and the rth succeeding term begin- 
ning with r number of noughts ; except indeed that the coef- 
ficients of the terms will sometimes have the order of signs 
interrupted, namely, when a greater value is to be subtracted 
from a less. 
But if every set should have the same order to signs con- 
trary to those in the set immediately preceding, and conse- 
quently every set omitting one set continually, have the same 
order of signs, then by continually multiplying by 2 cos. of 
we shall get new series by taking the differences of the 
coefficients of any term and the rth term from it. 
Scholium 11. We may by the methods above not only find 
the valuation of infinite series, but likewise of finite series. 
Example 1 , Required the sum of the r first terms of the series, 
cos. of nz- 1 - cos. of n-\-q.z-{- cos. of n^zq.z & c. 
The series ad infinitum may be written thus, cos. of nz- 1 - 
cos. of n-\-q.z-\- cos. of n-{-Qq.z -f- cos. of n+r—i.qz 
-f cos. of n-\-rq.z-\- cos. of n-\-r- j-i .q-%-\- &c. ad infinitum , 
from which if we take cos. of n-\-rq.z-\- cos. of n+r+i.q.z 
+ cos. of n+r+2.q.z &c. ad infinitum , we shall have the 
required sum ; the first of these by Example 2, Theorem I. 
= — , sine ° f ”-fc * and t h e second by the same, by merely 
2 sine or \qz 1 J J 
