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?