Series. 187 



hand side of the last inequality approaches zero ; therefore the 

 remainder after ri— i terras approaches zero : therefore the series 

 is convergent. 



8. In the theorems of the two preceding articles the student 

 should note the force of the words ''after a certahi number of 

 terms. ' ' The first few terms of a series may not give any indica- 

 tion as to whether the series is convergent or divergent. Take, 

 for example, the series 



where the rth term is rx''~\ and suppose -^t^yV ; then the succes- 



sive terms grow larger up to the ninth term, which =^3. The 



tenth term has the same value as the ninth, but every term after 

 the tenth is less than the preceding one. Moreover, as n increases 

 without limit, the n th term approaches zero and the ratio of the 



(7i-\- 1 jth term to the 7Hh term equals | i -f | x, and this evidently 



I ^^ J 

 approaches x as a limit. Hence the series is convergent. 



9. Theorem. A series is convcrgeyit if the series obtained by 

 7naking all its terms positive is convergent. 



Let the limit of the sum of the positive terms be represented by 

 U^. and the limit of the sum of the negative terms be represented 

 by Uj then the limit of the sum of the series will be U,— U.. 



Now consider a new series formed from the given series by 

 making all its terms positive ; then the limit of the sum of this 

 new series will be U^-f U^, and as this new series is convergent by 

 hypothesis, U,-f U,, has a definite value. Again, as U, and U^ are 

 both positive and as their sum has a definite value, therefore each 

 of these quantities U^ and U, has a definite value, therefore their 

 difference, U^~U„ has a definite value ; therefore the .series is 

 convergent. 



10. Theorem. The series 



a^-{-a^x-\-a^x^-\-a^x^-\- . . . 

 is convergent when x<^\., unless a„ increases without limit as n in- 

 creases ivithout limit. 



