186 Algebra. 



Moreover, 8^= 2;f i and S,,i2r differ by ii,^.^^,^^, which approaches 

 zero as r increases ; therefore the two quantities between which S 

 is always found approach equality as r increases. Therefore S 

 has a definite value, or, in other words, the series is convergent. 



7. Theorem. If all the terms of a series are positive and after a 

 eertain number of terms each term is less tha?i the one before it and 

 the limit of the nth term is zero ; then if the limit of the ratio of the 

 {n-\-\)th term to the nth term is less than i the series is convergent. 



If all the terms are positive and after a certain number of terms 

 each term is less than the preceding one, then, anywhere after 

 this certain number of terms, the ratio of any term to the preced- 

 ing one is positive and less than i. Now, since each of these 

 ratios is less than i and the limit of the ratio is less than i, we 

 may take some quantity, /', less than i but so near i that each 

 ratio will be less than k. 



. ; 



or, adding n„ to each side of the inequality, 



/y„ + /^„; ,+^^,, ,-[-?/, 13 4- . . . <uUi-\-k-hk'^-\-k-^-\- . . . ) 

 But when /i'<i 



i-f/j+Z'^-f /'••+ . . . = \ vSee XII, Art. 8. 



I— A- 



and this is a definite quantity. 



Therefore the right-hand side of the last inequality is a definite 

 multiple of n,„ and n„ ^ o ; therefore the right-hand side of the 

 last inequality approaches zero ; and as the left side is less than 

 the right side and -neither side can be negative, therefore the left- 



