I'm having a lot of trouble showing that that the
I figure it has to be done by induction, so if I wanted to show that tends to infinity faster then I would have:
which is obviously true
so assume that
is true
then I would have where the inequality holds only if tends to infinity at a faster rate then
p is a real number. so induction wouldn't work here! also why is this question in number theory section? your notation is not very good. you actually want to show that for sufficiently large values
of we have which will follow if you show that this is clear for for let and apply L'Hospital rule times to get the result.
I will be explicit in my hint, I guess you did not see what I was trying to say. Let and . Pick so that . If we can prove that for sufficiently large the proof will follow because . However, we know that . However, if we have . Thus, we have shown that eventually overtakes and so it overtakes .