Proving n^n > 2^n *n using the Binomial theorem

In summary, the conversation discusses proving the inequality n^n > 2^n * n! using the Binomial theorem. The person has attempted to use induction and converting the equation into the form of the Binomial theorem, but has not been successful.
  • #1
dot.hack
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0

Homework Statement


Prove that [itex] n^n > 2^n * n! [/itex] when n > 6 using the Binomial theorem.
I just proved the Binomial theorem using induction which was not that difficult but in applying what I learned through it's proof I am having difficulty.

Homework Equations


Binomial theorem = [itex] (x+y)^n = \sum_{k=0}^n\binom{n}{k}x^{n-k}y^k [/itex]


The Attempt at a Solution


I attempted setting n= (x+y) to convert the left side of the equation into the form of the binomial theorem, as well as turning the right hand side into the form of the binomial theorem by setting x+y = 2 both to no avail. Actually the "closest" (I put this in quotes because as I couldn't solve it, I have no idea how close I really was) I got was by using induction and turning the equation into [itex]{\frac{(n+1)^n }{2}}= 2^n + n![/itex]
Thanks for the help guys.
 
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  • #2
Actually, the last equation I wrote have n!/2
thanks
 

Related to Proving n^n > 2^n *n using the Binomial theorem

1. How is the Binomial theorem used to prove n^n > 2^n *n?

The Binomial theorem is a formula used to expand a binomial expression raised to any positive integer power. By applying this formula to the expression n^n, we can show that it is greater than 2^n *n.

2. What is the process for using the Binomial theorem to prove this inequality?

The first step is to expand the expression n^n using the Binomial theorem. Then, we can simplify the resulting terms and rearrange them to show that it is greater than 2^n *n. This process may involve using algebraic manipulations and properties of exponents.

3. Can the Binomial theorem be used to prove other inequalities?

Yes, the Binomial theorem can be used to prove various inequalities involving binomial expressions. It is a powerful tool in combinatorics and algebraic proofs.

4. Are there any limitations to using the Binomial theorem in proofs?

Yes, the Binomial theorem can only be applied to binomial expressions raised to positive integer powers. It cannot be used for expressions with negative exponents or fractional powers.

5. Are there alternative methods for proving n^n > 2^n *n?

Yes, there are other methods for proving this inequality, such as using mathematical induction or properties of logarithms. However, the Binomial theorem is a commonly used and efficient method for this particular proof.

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