184 AlvGKBRA. 



3. With the notation just explained, the sum of a series which 

 has a limited number of terms will be represented by S^ + R^, 

 whether ^ is i or 2 or 3 or any other number not exceeding the 

 whole number of terms of the series. 



In an infinite converge7it series S;, approaches a limit as n in- 

 creases without limit, and the value of this limit is 



S^ + R,^, 

 where q is any positive whole number whatever. It is easy to 

 see in this case that R^,^ o as n increases without limit. 



In an infinite diverge7it series S„ does not approach any limit 

 neither does R,, approach an}^ limit, and S^ + R;^ has no definite 

 value at all. 



4. It is evident that a series cannot be convergent unless after 

 a certain number of terms are taken the successive terms grow 

 smaller and smaller, or, in other words, unless ?/„^ o as 11 in- 

 creases without limit. But while this is necessary it is not suf- 

 cient, for a series may be divergent and still u,^ o as n increases 

 without limit. 



Take for example the series 



1+1+^+1+ 



where the ?^th term is , which evidently approaches zero as n 



increases without limit. 



If this series be grouped thus : 



i+-Ha+i) + (i+-,V+-K*) 



+ a + TV + TV + TV + A + T'4+TV + TV)+ . • ■ 

 then in no group is the sum less than \ and as there are an un- 

 limited number of groups, the series evidently does not approach 

 any limit, but increases wathout limit as the number of terms in- 

 creases without limit, therefore the series is divergent. 



5. Theorem. A series, all of whose ler?ns ai-e positive, is diver- 

 gent if nu,, does not approach zero as n ificreases ivithoiit limit. 



Since all the terms are positive, 7111^ is positive, and since 7iu„ 



does not approach zero, we may take r a quantity so near zero 



r 



that n7i„^r, then 7i„'>~-. 



n 



