vSkriks. 185 



r 



n-\-2 



etc. 



\n u-j-i H-f-2 ■ 



But the quantities in the parenthesis form the terms of the 

 series i+i + -^-f . . . after the nth term, and this latter series 

 has been shown to be divergent, or in other words, the quantity 

 in the parenthesis increases without limit," and therefore 



increases without limit ; therefore the series is divergent. 



6. Theorem. //' the terms of a series are alternately positive and 

 jiegative and after a certabi number of terms each term is numeri- 

 cally less than the preceding one, and the n th term approaches zero, 

 as 71 hicreases 7vlthout limit, the series is convergent. 



Let the series be 



u^ — u^-\-u^ — u^-\- . . . 

 and let the series be represented by S ; then we may write either 



S=S,-h(ii,+ i—u,+ 2) + (^^,+ s—^^,+4)-h ... (t) 



or S=S,/,i — r?Vf2 — ?^H J — r^^^+4 — ?Woy>— . • • (2) 



After a certain number of terms, say k, each term is less than 



the preceding one, so if (/ be larger than k, each parenthesis in 



(i) and also in (2) is positive, and therefore 



from (i) S>S^, 



and from (2) S<S^4 1. 



Thus we see that S is intemie'diate in value between the two 



definite quantities S^ and S^n, which two quantities diifer 



by n,^.^.. 



Similarly, whatever positive whole number be represented by 



r, we get 



and S<S,^,2r+i. 



Now Sy-f2r>S,/ 



and w5y -f 2 »-+ 1 ^^ v^^r-i I . 



Therefore S is intermediate in value between two quantities, 

 the larger of which grows smaller and the smaller of which grows 

 larger. 



