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#1 2014-01-24 05:34:23

Bezoux
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Fascinating limit

Hi, everyone.
I had an interesting problem in class today. Find

.
Now, it's fairly obvious that the answer is 0, the question is how to prove it.
I had an immediate suggestion, obviously if
,
then the answer is 0. So, we had to prove that.
By definition,

Since both 
and M are positive, I can raise them to the degree of n, which is when I get
, which is obviously correct for n>= a certain n0.
Now, my teacher told me this was wrong and the class ended before he could elaborate on why, and the test is next class, which worries me a bit. I do a lot of my proofs this way.
What she said is that we need to prove
, after which we can easily use the squeeze theorem to solve the problem. Unfortunately, I've had no success in proving that statement.
Can you point out where I was wrong in my solution, or help me prove the teacher's statement? I keep getting stuck at
, but the teacher doesn't want us to use e (or infinite geometric series, which would make the proof fairly straightforward by using the binomial formula).
Thank you in advance, I know I'm asking a lot.

Last edited by Bezoux (2014-01-24 05:35:27)

#2 2014-01-24 19:42:39

bobbym
Administrator

Online

Re: Fascinating limit

Hi;

You mean you can not use the well known fact that


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#3 2014-01-25 08:59:47

Bezoux
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Re: Fascinating limit

Yes.
Is there a way to prove that that sequence is always lesser than 3 without e?

#4 2014-01-26 02:20:00

bobbym
Administrator

Online

Re: Fascinating limit

Hi;

I am not getting anywhere with the restrictions that are placed on the problem. But starting with



perhaps you could replace the n! with Stirlings formula.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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