Prove the limit of the nth root of n

[Carla1985]

I don't understand what I have to do with this question. do i just explain what is happening in each part or is there more to it?

http://imageshack.us/scaled/medium/826/screenshot20130214at150.png
Thanks

 

[chisigma]

I don't understand what I have to do with this question. do i just explain what is happening in each part or is there more to it?

http://imageshack.us/scaled/medium/826/screenshot20130214at150.png
Thanks

The point (i) is justified by the series expansion... $\displaystyle (1+x)^{n}= \sum_{k=0}^{n} \binom {n}{k}\ x^{k}$ (1) ... where all terms are positive and $\displaystyle \binom{n}{2}\ x^{2}$ is the only second term. The other steps are quite simple... Kind regards$\chi$ $\sigma$

 

[MarkFL]

It appears that you are being asked to demonstrate that each step is correct.

For example, for (i) you could write:

$\displaystyle (1+x)^n=\sum_{k=0}^n{n \choose k}x^k={n \choose 2}x^2+\left(1+nx+\sum_{k=3}^n{n \choose k}x^k \right)$

For $k\ge0$ it must be true that:

$\displaystyle 1+nx+\sum_{k=3}^n{n \choose k}x^k\ge0$

and so by adding $\displaystyle {n \choose 2}x^2$ to both sides, we obtain:

$\displaystyle (1+x)^n\ge{n \choose 2}x^2=\frac{n(n-1)}{2}x^2$

For (ii) you could begin with:

$\displaystyle \frac{n}{2}\ge1$ for $n\ge2$

Now add $\displaystyle \frac{n}{2}$ to both sides:

$\displaystyle n\ge1+\frac{n}{2}$

Subtract through by 1:

$\displaystyle n-1\ge\frac{n}{2}$

Now, multiply through by $\displaystyle \frac{n}{2}$

$\displaystyle \frac{n(n-1)}{2}\ge\frac{n^2}{4}$

From (i) we have:

$\displaystyle (1+x)^n\ge\frac{n(n-1)}{2}x^2$

And so on the right, replacing $\displaystyle \frac{n(n-1)}{2}$ with $\displaystyle \frac{n^2}{4}$ we obtain:

$\displaystyle (1+x)^n\ge\frac{n^4}{4}x^2$

Multiplying through by $\displaystyle \frac{4}{n^2}$ there results:

$\displaystyle \frac{4}{n^2}(1+x)^n\ge x^2$

$\displaystyle x^2\le\frac{4}{n^2}(1+x)^n$ where $n\ge2,\,x\ge0$

Now for (iii), it is just a matter of applying the result of Problem 1(ii) (which you haven't provided).

 

[Carla1985]

Ah, I see, thankyou. Q1 ii is to prove that limit 1/sqrt(n)=0

I'm assuming that we do that using the definition of convergence and work out an N(E), E being epsilon, would that be right?

 

[MarkFL]

I think I would rewrite the problem first before using an epsilon-delta argument:

$\displaystyle \lim_{n\to\infty}\frac{1}{\sqrt{n}}=0$

$\displaystyle \lim_{n\to\infty}\sqrt{\frac{1}{n}}=0$

$\displaystyle \sqrt{\lim_{n\to\infty}\frac{1}{n}}=0$

$\displaystyle \lim_{n\to\infty}\frac{1}{n}=0$

 

[Carla1985]

I didnt know I could do that, that may help in future sums :) I was just making sure I was doing it in the right way, we get given so many definitions and corollaries etc :)

 

[Sudharaka]

I don't understand what I have to do with this question. do i just explain what is happening in each part or is there more to it?

http://imageshack.us/scaled/medium/826/screenshot20130214at150.png
Thanks

Hi Carla1985, :)

Thought you might be interested in seeing http://planetmath.org/LimitOfNthRootOfN.html.

Kind Regards,
Sudharaka.