Maths proof?

Question

How can you prove that there exists a real number c independent of n such that

a^n >= c*n^(k+1)

where a is a positive real, k is a positive integer, and c is independent of n, so that it is true for n=1,2,3,...

I've tried letting a=1+b and using the binomial expansion and it seems to help, but i'm still getting stuck. i think it will be to do with sequences and limits etc as well.

thanks

Answer 1

If 0 If a>1 lim a^n/n^(k+1) = + infinity so it becomes greater as any positive arbitrary number c ,but given c you can find No so for any n>No
a^n> c*n^(k+1) but this is not true for n=1,2,3.. only for all those n>No where No depends on c
So your statement is NOT true