Theory of Limits. 151 



2. Find the limit of the progression i + i-f-i4-TV+ ^^^-^ ^^ 

 n increases. 



J. Find the limit of .272727+ as n increases. 

 ^. Find the limit of .279279279+ as n increases. 



5. Find the limit of the sum of i— i + TV~2V+ ^t^-- ^^ ^ 

 increases. 



6. Find the limit of the sum of V 8+ V 4+ V 2+ V i -f 

 as 71 increases. 



25. Theorem. The limit of the siim of the series 1+/+;-' + 

 ^'3_j_,-4_^ ^/r., as r decreases and n increases is i. 



In the equation 



lim ^= 



I— r 



the expression will of course have different values for different 

 I — r 



values of r. Hence we may, if we choose, look upon this expres- 

 sion as a variable. But as r approaches o as a limit the fraction 



approaches « as a limit. Therefore we may say that in a decreas- 

 ing geometrical progression as the number of terms increases with- 

 out limit, and as the ratio approaches zero as a limit, the sum 

 approaches a as a limit. 



In particular, then, if ^=1 and if the number of terms in- 

 creases without limit, and the ratio approaches zero as a limit, the 

 series 



1+/-+/"+ . . .• 

 approaches i as a limit. 



26. Theorem. The limit of the series 



A, + A,.r+A^.r= + A^.r3+ . . . , 

 as the number of terms increases iL'ithont limit and as x approaches 

 zero, is A^. 



Take the series first without the A ,, and suppose K to be a 

 positive quantity numerically equal to the greatest of the co- 

 efficients A,, A^, A^, . . . Then 



A.v+A,v^+A,r^+ . . . numerically<K('x+A--' + .r^+ . .; 



