2009/014) sum of the reciprocals 1+1/2+1/3+...+1/n is never an integer for n >1

prove that the sum of the reciprocals 1+1/2+1/3+...+1/n is never an integer for n >1


let S = 1+1/2+1/3+...+1/n

because n > 1 so from 1 to n there is only one number 2^k such that

2^k <= n < 2^(k+1) that is the higest power of 2 less than <=n

2^k devides that number (as 2^k devides 2^k) and no other number

now take LCM(2,3,...n ) = p and mutlipy S by p

now 2^k devides p
each term of RHS except 1/2^k as denominator term is not divisble by 2^k gives an even number but 1/2^k gives an odd number p is not divisible by 2^(k+1)

so RHS = odd

pS = odd

so S = odd/p

p is even

so S = odd/even and hence not an integer