34 BOOK OF MATHEMATICAL PROBLEMS. 



177. The coefficient of x r in the expansion of (1 + x) m being 

 denoted by a r , prove that 



p being a whole number < n. 



178. Denoting by u n the series 



1- + 2" + jg + ... + *' ' '/ + ... to infinity, 

 prove that 



and w,, + 1 - w w = 



l) n(n-l)(n-2 



~ 



[2 - 8 [3 .-3... 



and by means of either of these, prove that w 7 = 4140c. 



179. If u n = I"' 1 -2- 1 + ~ - ^+ ... to infinity, 



then will 



w(w 1) 





then will 



181. Having given 



