Can someone give me a hint as to where to start while attempting this problem?
Use mathematical induction to prove that n!<n^n whenever n is a positive integer greater than 1.
Multiply both sides of:
by to get:
So:
Now taking the first two terms of the binomial expansion of we get:
where is the remainder (which is greater than zero because all of the terms left out of this are also positive).
So:
Hence:
and so going back to we have:
which completes the proof of the induction step.
RonL
[quote=CaptainBlack;164709]Multiply both sides of:
by to get:
So:
Now taking the first two terms of the binomial expansion of we get:
Can you please explain how to factor these, I don't understand how, for example (n+1)Xn^n becomes n^n+1+n^n. I also am not understanding the part where it says taking the first two terms of the binomail expansion... where did that come from?
I'm sorry to sound so crazy, I just am not grasping this quite yet!