3 20 NEW SERIES for the 



52. Let v denote any number greater than unity, and p and 

 n any two whole pofitive numbers ; then, by a known formula 



v p — I = V M v -f- v z + v 3 -f v* . . . + v p >, 



v n — 1 — V * \ v + v % -f- <i> 3 -f V* . . . + v* J- j 



therefore, dividing each fide of the firft of thefe equations by 

 the correfponding fide of the fecond, we get 



v p — 1 _ v + v 1 -j- v* -f- V 4 . . . -f- v p 

 v n — 1 ~~ v -f v -f- v* + i> + . . . + v n * 



Now, v being by hypothefis greater than unity, the fraction 

 on the right hand fide of this equation is lefs than this other 

 fraction 



v p -\- v p + v p + v p . . . + v p (to p terms) _ p v p 

 i-{-i-f-i + i...+ i(to/z terms) ' ' ~' 



becaufe it has manifeftly a lefs numerator, and at the fame time 

 a greater denominator. The fame fraction is, however, greater 

 than this fraction 



1 -f- T + 1 + 1 •»•+ 1 (to p terms) __ p 



v n _f_ v » _}- V » -\- v n . . . + i) n (to n terms) ~"~ n v n ' 



becaufe it has a greater numerator, and a lefs denominator. 

 Therefore, 



qjP J p V P V P I p 



v" — 1 < n ' V 1 — 1 > „ v ** 



and hence, dividing the firft of thefe expreffions by vp, and mul- 

 tiplying the fecond by v", 



> aJfaSI TV Z < „,* __ X * W' 



n i^(w s — i)' n v n — 1 



53. Putting 



