You are Here: Home >< Maths

# Inequality watch

1. Hey, trying to prove that there exists an epsilon in (0,e) such that there exists an N in N so the following holds;

for all n > N.

I think it is true (noting that (n/3)^n < n! < (n/2)^n) and then abusing stirlings formula a bit to obtain no contradictions if that was a case.

However, I am finding it hard to prove that it does exists.
2. I'm not sure but I would rearrange to get

letting

Now, think of it as a sequence. Does it converge?
3. That dosen't really do alot as far as I can tell because convergence still depends upon the value of a - for instance if a = 1/2 we have divergence and if a <= 1/3 we have convergence. I just choose 1/(e-epsilon) because of these facts but deducing anything about the behaviour with such values of a is proving hard.
4. (Original post by DeanK22)
That dosen't really do alot as far as I can tell because convergence still depends upon the value of a - for instance if a = 1/2 we have divergence and if a <= 1/3 we have convergence. I just choose 1/(e-epsilon) because of these facts but deducing anything about the behaviour with such values of a is proving hard.
Use the ratio lemma? To find exactly when it changes from convergence to diverge BECAUSE of a change in a.

Spoiler:
Show

If ea>1 then it's gonna diverge. This happens precisely when a>e^-1, and since a is going to be bigger than e^-1
5. Merci, never come across that lemma before but it is pretty useful!

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: January 19, 2010
Today on TSR

### He lied about his age

Thought he was 19... really he's 14

### University open days

Wed, 25 Jul '18
2. University of Buckingham
Wed, 25 Jul '18
3. Bournemouth University
Wed, 1 Aug '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams