Fascinating limit

Bezoux · 2014-01-23 06:34:23

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 (2014-01-23 06:35:27)

bobbym · 2014-01-23 20:42:39

Hi;

You mean you can not use the well known fact that

Bezoux · 2014-01-24 09:59:47

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

bobbym · 2014-01-25 03:20:00

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.

Math Is Fun Forum

#1 2014-01-23 06:34:23

Fascinating limit

#2 2014-01-23 20:42:39

Re: Fascinating limit

#3 2014-01-24 09:59:47

Re: Fascinating limit

#4 2014-01-25 03:20:00

Re: Fascinating limit

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