Limits

**mathsohard** · January 20th 2011, 05:35 PM

1. Show that lim (n^n)/(n!e^n) exists without finding the limit.
n->oo

2. Show that lim (1/n){(2*4*...*(2n))/(1*3*...*(2n-1))}^2 exists without finding the limit. n->oo

I guess I have to prove this by least upper bounds, however hard to start!

**dwsmith** · January 20th 2011, 05:38 PM

Originally Posted by mathsohard

1. Show that lim (n^n)/(n!e^n) exists without finding the limit.
n->oo

2. Show that lim (1/n){(2*4*...*(2n))/(1*3*...*(2n-1))}^2 exists without finding the limit. n->oo

I guess I have to prove this by least upper bounds, however hard to start!

For 1, the denominator grows faster than the numerator. Hence, the limit is 0.

**tonio** · January 20th 2011, 06:56 PM

Originally Posted by dwsmith

For 1, the denominator grows faster than the numerator. Hence, the limit is 0.

The same could be said about $\displaystyle{\frac{n}{n+1}}$ , yet this seq. does not converge to zero...

Tonio

**dwsmith** · January 20th 2011, 07:00 PM

Originally Posted by tonio

The same could be said about $\displaystyle{\frac{n}{n+1}}$ , yet this seq. does not converge to zero...

Tonio

Limits of a simple polynomial we just evaluate the fraction of the highest power variable. I would have changed my answer to take n/n then but for this case and since it doesn't say solve the limit, I gave the poster a correct answer for his limit.

**tonio** · January 20th 2011, 07:12 PM

Originally Posted by mathsohard

1. Show that lim (n^n)/(n!e^n) exists without finding the limit.
n->oo

Show that your sequence is monotone descending, which is surprisingly easy if you already know that the sequence

$\displaystyle{\left(1+\frac{1}{n}\right)^n$ converges monotonically ascending to $e$

2. Show that lim (1/n){(2*4*...*(2n))/(1*3*...*(2n-1))}^2 exists without finding the limit. n->oo

I guess I have to prove this by least upper bounds, however hard to start!

You can try here the above, but it'll be much more involved. First, be sure you can show that

$\displaystyle{2\cdot 4\cdot\ldots\cdot (2n)=2^n\,n!\,,\,\,1\cdot 3\cdot\ldots\cdot (2n-1)=\frac{(2n)!}{2\cdot 4\cdot\ldots\cdot (2n)}}$ , so

that now you can write your sequence as $\displaystye{a_n:=\frac{1}{n}\left(\frac{(2^n\,n!) ^2}{(2n)!}\right)^2$ , and

now show that $a_{n+1}\leq a_n$ ...very carefully and slowly!

Tonio

.

**tonio** · January 20th 2011, 07:16 PM

Originally Posted by dwsmith

Limits of a simple polynomial we just evaluate the fraction of the highest power variable. I would have changed my answer to take n/n then but for this case and since it doesn't say solve the limit, I gave the poster a correct answer for his limit.

I'm not trying to start an argument here but your answer isn't correct since my example contradicts it. You didn't explain anything

about polynomials, and even if you did that doesn't prove that a sequence which is given in form of a fraction will

necessarily converge to zero if simply, as you wrote, "the denominator grows faster than the numerator".

Tonio

**Drexel28** · January 20th 2011, 08:10 PM

Originally Posted by mathsohard

1. Show that lim (n^n)/(n!e^n) exists without finding the limit.
n->oo

2. Show that lim (1/n){(2*4*...*(2n))/(1*3*...*(2n-1))}^2 exists without finding the limit. n->oo

I guess I have to prove this by least upper bounds, however hard to start!

Try using the ratio test.

**mathsohard** · January 20th 2011, 10:59 PM

Originally Posted by tonio

.

To show it is a monotone descending sequence, I just have to show that n < n+1 right?? and then how can I prove it converges???
need little more help on both of them

**tonio** · January 21st 2011, 05:50 AM

Originally Posted by mathsohard

To show it is a monotone descending sequence, I just have to show that n < n+1 right?? and then how can I prove it converges???
need little more help on both of them

Well, as both sequences are obviously positive, showing they are monotonee descending automatically

makes them monotone and bounded and thus converging.

Tonio

**HallsofIvy** · January 21st 2011, 06:04 AM

Originally Posted by mathsohard

To show it is a monotone descending sequence, I just have to show that n < n+1 right?? and then how can I prove it converges???
need little more help on both of them

To show that a sequence is monotone ascending you would need to show that $a_n< a_{n+1}$ . To show a sequence is monotone descending, you would need to show that $a_n> a_{n+1}$ .

If a monotone ascending sequence has an upper bound, then it converges. If a monotone descending sequence has a lower bound, then it converges. Sequences of positive numbers have, by definition of "positive", 0 as a lower bound.

**mathsohard** · January 21st 2011, 12:12 PM

I have another question, I understand that if a sequence is monotone ascending and have an upper bound, there is a limit, and if monotone descending and have a lower bound there is a limit. And to show a sequence is ascending I have to show An < An+1 and for descending An > An+1. But for this problem n^n/(n!e^e), I know it is descending and has lower bound of 0 since it is positive, however what is the proper way of showing that An > An+1 ??? I can't think of anything else but plugging in...

**tonio** · January 21st 2011, 02:16 PM

Originally Posted by mathsohard

I have another question, I understand that if a sequence is monotone ascending and have an upper bound, there is a limit, and if monotone descending and have a lower bound there is a limit. And to show a sequence is ascending I have to show An < An+1 and for descending An > An+1. But for this problem n^n/(n!e^e), I know it is descending and has lower bound of 0 since it is positive, however what is the proper way of showing that An > An+1 ??? I can't think of anything else but plugging in...

As in my last post, if you put $\displaystyle{a_n:=\frac{n^n}{n!\,e^n}}$ , then we get

$\displaystyle{a_{n+1}\leq a_n\Longleftrightarrow \frac{(n+1)^{n+1}}{(n+1)!\,e^{n+1}}\leq \frac{n^n}{n!\,e^n}}\Longleftrightarrow \frac{(n+1)^n}{n^n}\leq e}$ , and we get

that this last inequality is true since, as was observed in my past message, the left side

in it converges monotonically ascending to the right side.

Tonio

**Hartlw** · January 22nd 2011, 11:31 AM

It seems to me if you have to show something converges without finding a limit then Cauchy's convergence theorem is worth a try.

**Hartlw** · January 24th 2011, 03:50 AM

Use ratio test proposed by Drexel28 above:

An/An+1= (n+1)!/n! X n^n/(n+1)^(n+1) X e^(n+1)/e^n

= (n+1/n)! X (n/n+1)^n X 1/(n+1) X e^(n+1)/e^n

= (n+1/n)^n X e > 1

An+1/An < 1 => monotonically decreaing, and > 0.

(Sorry about Cauchy. Didn't read everything and assumed ratio test had been tried.)

**DrSteve** · January 24th 2011, 04:29 AM

Originally Posted by tonio

The same could be said about $\displaystyle{\frac{n}{n+1}}$ , yet this seq. does not converge to zero...

Tonio

I wouldn't say that $n+1$ grows faster than $n$ . I would say that $n+1$ is greater than $n$ for all $n$ , but that they grow at the same rate.

I'm not saying that this is wrong, but maybe a better counterexample would be something like $\frac{n^2}{n^2+n}$ .

I also think that dwsmith's statement is correct if we add the word "much." That is, the denominator grows "much faster" than the numerator (in the original question). When I say that a function $f$ grows much faster than a function $g$ , by that I mean $\lim_{x\rightarrow \infty\frac{g(x)}{f(x)}=0$ . Although I suppose this is just rephrasing the original question.

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