CHAPTER XIV. 



SERIES. 



1. A definition of a series was given in XII, Art. 4, and it was 

 there noticed that infinite series are divided into the two classes 

 of convergent and divergent. Convergent series have definite 

 limits as the number of terms is increased without limit, but from 

 their nature divergent series are wholly indefinite, and hence // is 

 not safe to use divergent series or to base any reasoning upon them. 



In all that follows, and indeed in all that precedes, it is to be 

 understood wherever infinite series are used that the results hold 

 as long as all the series used or obtained are convergent. 



In many cases a series is convergent or divergent according to 

 the value of some letter in the series, and it is always understood 

 in such cases that the letter concerned is limited to tho.se values 

 which make the series convergent, and no inference is to be 

 drawm for any other value. 



It would be fortunate if some simple and universal criterion 

 were known whereb}- we might determine whether any given 

 series is convergent or divergent, but unfortunately no such cri- 

 terion has been found. There are, however, some cases in which 

 we can determine whether a .series is convergent or divergent and 

 we give a few of these. 



2, L-t the terms of a series be represented b}^ u^, u^, u,, etc., 

 in each case the subscript being the same as the number of the 

 term; and let R^ be the remainder after the first term, R^ the re- 

 mainder after the second term, R, the remainder after the third 

 term, etc. ; in each case the remainder after any number of terms 

 are taken is represented bj^ R with a subscript equal to the num- 

 ber of terms already taken; and further let the sum of any number 

 of terms be represented by S with a subscript equal to the num- 

 ber of terms taken, /. e. the sum of two terms will be represented 

 bv S., the sum of three terms bv S,, and so on. 



