796] 
617 
796. 
SERIES. 
[From the Encyclopcedia Britannica, Ninth Edition, voi. xxi. (1886), pp. 677—682.] 
A series is a set of terms considered as arranged in order. Usually the terms 
are or represent numerical magnitudes, and we are concerned with the sum of the 
series. The number of terms may be limited or without limit; and we have thus the 
two theories, finite series and infinite series. The notions of convergency and divergency 
present themselves only in the latter theory. 
Finite Series. 
1. Taking the terms to be numerical magnitudes, or say numbers, if there be a 
definite number of terms, then the sum of the series is nothing else than the number 
obtained by the-addition of the terms; e.g. 4 + 9 + 10 = 23, 1+2 + 4 + 8=15. In the 
first example there is no apparent law for the successive terms ; in the second example 
there is an apparent law. But it is important to notice that in neither case is there 
a determinate law : we can in an infinity of ways form series beginning with the 
apparently irregular succession of terms 4, 9, 10, or with the apparently regular 
succession of terms 1, 2, 4, 8. For instance, in the latter case we may have a series with 
the general term 2 W , when for n = 0, 1, 2, 3, 4, 5,... the series will be 1, 2, 4, 8, 16, 32,...; 
or a series with the general term ^ (w 3 + 5w + 6), where for the same values of n the 
series will be 1, 2, 4, 8, 15, 26,... The series may contain negative terms, and in 
forming the sum each term is of course to be taken with the proper sign. 
2. But we may have a given law, such as either of those just mentioned, and 
the question then arises, to find the sum of an indefinite number of terms, or say of 
n terms (n standing for any positive integer number at pleasure) of the series. The 
expression for the sum cannot in this case be obtained by actual addition ; the 
formation by addition of the sum of two terms, of three terms, &c., will, it may be, 
suggest (but it cannot do more than suggest) the expression for the sum of n terms 
c. xi.