1 88 A1.GKBRA. 



We may consider all the terms positive, for if some were nega- 

 tive we could form a new series all of whose terms were positive 

 and conduct the reasoning upon the new series, and if this new 

 series were convergent the original series would be so by Art. 9. 



Since we may consider all the coefficients positive and since a„ 

 does not increase without limit we may take b, a quantity greater 

 than the greatest of the coefficients, then 



a^^-\-a^x-\-a^x^-\- . . . <,d-j-dx-\-dx'~-{- ... 



But the right-hand side of this inequality equals ,(see XII 



I — X 



Art. 8,) /. e. the right-hand side has a definite value, therefore 



the left-hand side also has a definite value ; hence the series i-s 



convergent. 



Ij, Theorem, //'we have given a series such that after a certain 

 number of terms each term is less than the correspondi7ig tenn 0/ 

 some scf ics luhich is known to be convergent, then the giveyi series is 

 convergent. 



Let the given series be 



u^-^u^-\-u^-\-u^-\- . . , (l) 



and let the series known to be convergent be 



^+^', + ^•34-^'^+ ... (2) 



and suppose each term after the rth in (i) to be less than the cor- 

 responding term in (2). 



Since (2) is convergent, 



approaches a definite limit as the number of terms increases with- 

 out limit, and since ?/, , ^ <t'..| -i, //;--] 2 <?'r!2> ^^r 1 s ^" '^V+ 3 > ^tc, 

 therefore ?^,.^i +z^^_i.2 + ^^r+;? + 2^;-+4+ . . . 



approaches a definite limit as the number of terms increases with- 

 out limit. Now as the sum of the first r terms of (i) is a definite 

 quantity and the sum of the terms after the rth approaches a def- 

 inite limit, it follows that the whole series 



Ji -{-ti -\-u -\-ii -\- . . . 

 I ' 2 ' 3 ' 4 ' 



approaches a definite limit as the number of terms increases with- 

 out limit, or in other words, the series is convergent. 



