Yes , expanding is a proper proof , or we can set and suppose is a power of a prime .
Then
The LHS from the inequality is while the RHS is by combining them we can see what we need to show :
which is obviously true for .
s(n) denotes the sum of divisors function
suppose n=p^k, where p is prime and k is a positive integer
s(p^k)= [(p^k+1)-1]/(p-1)
show:
1+ 1/p < s(n)/n < 1+ 1/(p-1)
Also if p=3 how large does k have to be to make s(n)/n > 1.4999
I know that if the above is expanded then it makes sense that each part is less than the next but I'm not sure if this is a proper proof as I am using what they have given and just expanding it and showing it's true. Is there another way to go about it? and I know k has to be something like 15, but is there an easier way to figure it out rather than just starting from k=1 and working my way up?