CHAPTER VIII. 
THE CONVERGENCE OF INFINITE SERIES AND 
INFINITE INTEGRALS. 
148. In Ch. IV we explained what was meant by saying 
that an infinite series is convergent, divergent, or oscillatory, and 
illustrated our definitions by a few simple examples, mainly 
derived from the geometrical series 
1 + x 4- ¿c 2 + ... 
and other series closely connected with it. In this chapter we 
shall pursue the subject in a more systematic manner, and prove 
a number of theorems which enable us to determine when the 
simplest series which commonly occur in analysis are convergent. 
149. Series of Positive Terms. We pointed out in Ch, IV 
(§ 10) that the easiest type of series to consider from the point 
of view of its convergence is that in which all the terms are 
positive (using positive to include zero). We shall consider such 
series first, not only because they are the easiest to deal with, but 
also because the discussion of the convergence of a series con 
taining negative or complex terms can often be made to depend 
upon a similar discussion for a series of positive terms only. 
When we are discussing the convergence or divergence of any 
series we may disregard any finite number of terms. Thus if a 
series contains a finite number only of negative or complex terms, 
we may omit them, and apply the theorems which follow to the 
remainder. 
150. It will be well to recall the following fundamental 
theorems established in § 70. 
A. A series of positive terms must he convergent or diverge 
to + oo , and cannot oscillate.