Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

You are not logged in.

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

Bezoux
Member
Registered: 2013-11-17
Posts: 7

### 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
,
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).

Last edited by Bezoux (2014-01-23 06:35:27)

Offline

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### 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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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

Bezoux
Member
Registered: 2013-11-17
Posts: 7

### Re: Fascinating limit

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

Offline

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

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### 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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline