322 NEW SERIES for the 



y y 



n t r — in r — i ' 



n 



r 



and hence, multiplying by n, and dividing by m t 



y_ y_ 



L > JL . n ( r *— *) L < r . * ( r * ~ *\ 

 m I 'm (r — i)' m m (r — i)' 



But ry — x, and r m — b, (Art. 51.)) from which it follows, that 



n n 



y _ 



r zz x , and r — b ', moreover, ~ == log x$ therefore, fubftitu- 

 ting, we get 



n {x n — 1) 1 n i. x " — 1) 



log x > -L . j , log x < b m . F , 



x m(b — 1) m [b — 1) 



and in thefe expreffions n denotes any whole v pofitive number 

 whatever. 



55. By fubtracting the lefTer of thefe limits to the logarithm 

 of x from the greater, we find their difference to be 



1 



n 



,m n 



n(x — 1) 

 4- 1 X (b x —1). 



n f .m - 



x m{b — 1) 



Now 



