This is what I have so far...
Basis step: P(2) states that 2! is less than 2^2. This is true since 2! = 2 and 2^2=4 and 2<4.
Inductive step: Assume n is greater than 1. Assume P(n) is true: that is
n! < n^n
This is where I am stuck!
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!