Expand & Simplify Binomial (1 + $\sqrt\frac{2}{n-1}$)^n

In summary, the conversation discusses expanding and simplifying a binomial expression involving a radical and finding the maximum value of n^\frac{1}{n}. The use of L'Hospital's Rule to prove a limit is also mentioned. The conversation then shifts to proving a inequality involving n^\frac{1}{n} and \sqrt\frac{2}{n-1}. The use of combinatorics in the proof is suggested. Finally, the question of how to prove that \sqrt{\frac{2}{n-1}} decreases slower than n^{{\frac{1}{n}}}-1 as n increases is brought up.
  • #1
recon
401
1
How do you expand and simplify [tex](1 + \sqrt\frac{2}{n-1})^n[/tex]?

I know this involves a binomial expansion and I can expand it to look something like

[tex]\left(\begin{array}{c}n&0\end{array}\right){\frac{2}{n-1}}^\frac{0}{2} + \left(\begin{array}{c}n&1\end{array}\right){\frac{2}{n-1}}^\frac{1}{2} + ...[/tex]

but how do you simplify this?
 
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  • #2
Sorry to dissapoint you,but apparently u cannot.It looks kinda ugly,but that's how it was supposed to be,since it involved a radical.

Daniel.
 
  • #3
Hmm. How can I solve the following problem then?

n is a positive integer. Prove that:

[tex]n^\frac{1}{n} < 1 + \sqrt\frac{2}{n-1}[/tex]

I also need to show that [tex]n^\frac{1}{n} \rightarrow 1 [/tex] as [tex]n \rightarrow \infty[/tex]. I know this follows logically from the fact that [tex]\frac{1}{n} \rightarrow 0 [/tex]as [tex]n \rightarrow \infty[/tex]. Is there a more rigorous way for showing this?

Also, what is the maximum value of [tex]n^\frac{1}{n} [/tex]?
 
  • #4
Do you know calculus??If u did,then
[tex] \lim_{n\rightarrow +\infty} n^{\frac{1}{n}}=\alpha [/tex](1)
U need to show that \alpha=1.
Take natural logarithm from both sides.Then
[tex] \lim_{n\rightarrow +\infty} \frac{1}{n}\ln n =\ln\alpha [/tex] (2)

The first limit is zero (you can show that considering the function [\itex] \frac{\ln x}{x} [/itex] and using L'Ho^spital rule.
THerefore [itex]\ln\alpha=0 \Rightarrow \alpha=1[/itex].

Daniel.
 
  • #5
I've never studied L'Hospital's Rule before (I just finished Grade 10). However, I just looked it up on the internet, and I do understand how it works, but not why it works.

Is this problem solvable?

If n is a positive integer, prove that:
[tex]n^\frac{1}{n} < 1 + \sqrt\frac{2}{n-1}[/tex]
 
  • #6
recon said:
I've never studied L'Hospital's Rule before (I just finished Grade 10). However, I just looked it up on the internet, and I do understand how it works, but not why it works.
L'Hospitals rule is relatively easy to prove using the definitions of limit and derivative.

Is this problem solvable?

If n is a positive integer, prove that:
[tex]n^\frac{1}{n} < 1 + \sqrt\frac{2}{n-1}[/tex]
If n=1 then there are some problems with this.
For n bigger than 1, you've almost got the proof.

Here's something you might find useful:
[tex]\left(\begin{array}{c}a&b\end{array}\right) = \frac{a!}{b!(a-b)!}[/tex]
Specifically
[tex]\left(\begin{array}{c}n&0\end{array}\right) = 1[/tex]
[tex]\left(\begin{array}{c}n&1\end{array}\right) = n[/tex]
and
[tex]\left(\begin{array}{c}n&2\end{array}\left) = \frac{(n-1)(n-2)}{2}}[/tex]
 
  • #7
NateTG said:
If n=1 then there are some problems with this.
For n bigger than 1, you've almost got the proof.

Here's something you might find useful:
[tex]\left(\begin{array}{c}a&b\end{array}\right) = \frac{a!}{b!(a-b)!}[/tex]
Specifically
[tex]\left(\begin{array}{c}n&0\end{array}\right) = 1[/tex]
[tex]\left(\begin{array}{c}n&1\end{array}\right) = n[/tex]
and
[tex]\left(\begin{array}{c}n&2\end{array}\left) = \frac{(n-1)(n-2)}{2}}[/tex]

Do you mean that I have to expand [tex](1 + \sqrt\frac{2}{n-1})^n[/tex]? It's the square root that is confusing me. I can't get rid of it. :yuck:
 
  • #8
Putting the question in another form, how do I proof that [tex]\sqrt{\frac{2}{n-1}}[/tex] decreases in value slower than [tex]{n^{{\frac{1}{n}}}-1[/tex] as n increases?
 

1. What is the purpose of expanding and simplifying a binomial expression?

Expanding and simplifying a binomial expression allows us to rewrite the expression in a more simplified form, making it easier to work with and solve. It also helps us to identify patterns and relationships between the terms in the expression.

2. How do you expand and simplify a binomial expression?

To expand and simplify a binomial expression, we use the binomial theorem or the FOIL method. We distribute the exponent to each term in the expression and then combine like terms to simplify the expression.

3. Can you give an example of expanding and simplifying a binomial expression?

For example, the binomial expression (x + 2)^3 would be expanded to x^3 + 6x^2 + 12x + 8, and then further simplified to x^3 + 6x^2 + 12x + 8.

4. What is the significance of the term (1 + $\sqrt\frac{2}{n-1}$) in the given binomial expression?

The term (1 + $\sqrt\frac{2}{n-1}$) represents the two possible solutions or roots of the expression, which are 1 + $\sqrt\frac{2}{n-1}$ and 1 - $\sqrt\frac{2}{n-1}$. These roots play a significant role in finding the values of variables in the expression and solving for the equation.

5. In what real-world situations would expanding and simplifying a binomial expression be useful?

Expanding and simplifying binomial expressions are useful in various fields of science, such as physics, chemistry, and biology. They can be used to simplify equations in motion and force, chemical reactions, and population growth models, among others. They are also essential in solving problems involving probability and statistics.

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