- #1
mr.tea
- 102
- 12
Homework Statement
prove: [tex] \lim_{n\rightarrow \infty} {\frac{n!}{2^n}}=\infty[/tex]
Homework Equations
Def. of a limit
The Attempt at a Solution
I would like to know if my solution is right or not. I think it is right but I would like to get a feedback. Please do not give me the answer, just directions/hints/things to think about, etc.
I need to show that for every (let's assume) positive natural number K (because I am going positively large, so negative numbers are not reasonable), I need to show that the sequence is larger then K from some place.
I thought to use the K I will be given to prove it. So I tried to find when:
[tex] \frac{K!}{2^K}>K [/tex]
and I have: [tex]\leftrightarrow K!>K*2^K \leftrightarrow (K-1)!>2^K [/tex]
And this is true for all K>=6. The proof is by induction(please feel free to correct me if I am wrong).
So, we take the max(K,7) and then it is true.
What do you think? (This is just a draft, so it will be more formal).
Thanks
Thomas