PRELIMINARY DEFINITIONS AND THEOREMS 
81 
9. The following theorem often affords a convenient means of 
estimating the remainder of an absolutely convergent series. 
Let A = aj + «2 + ••• be an absolutely convergent series. Let 
B = b 1 -\- b 2 + ■■■ be a positive term convergent series whose sum is 
known either exactly or approximately. Then if \a n \ < b n , n > m 
\A n \<B n <B. 
|-^n, p| ^**>1 + 1 T """ T ^n+p 
For 
< b n+l + *•• + b n+p 
<B n <B. 
Letting p= co gives the theorem. 
EXAMPLES 
81. 1. The geometric series is defined by 
G- = l+g+g‘ i +gZ^ 
(1 
The geometric series is absolutely convergent when \g|<1 and di 
vergent when |^|>1. When convergent, 
(2 
When g 1, 
= 1 + g + g 2 + ••• + g n 1 + -— 
1-g J 1-g 
Hence 
When |<jr|< 1, lim g n = 0, and then 
lim G- n = ---. 
1 -9 
When \g\ >1, lim g n is not 0, and hence by 80, 3, G- is not conver 
gent. 
2. The series 
(3