Prove monotonicity

1. Dec 4, 2012

hedipaldi

1. The problem statement, all variables and given/known data

prove that the sequence (1+1/n)^(n+1) is monotonic decreasing

2. Relevant equations

3. The attempt at a solution
I tried to use the binom expantion and the identity( Ck-1,n)+(Ck,n)=(Ck,n+1)

2. Dec 4, 2012

happysauce

All you have to do is show that a$_{n}$ ≥ a$_{n+1}$

3. Dec 4, 2012

hedipaldi

obviously,how do i do that?

4. Dec 4, 2012

When in doubt, use induction.

5. Dec 4, 2012

hedipaldi

If someone has an idea i will be glad to be helped.

6. Dec 4, 2012

I'm serious. Use induction. Prove first that (1+1/1)^(1+1) is greater than or equal to (1+1/2)^(2+1) as your base case.

I'm thinking about this problem and I think there actually is a better way of doing it but induction would be my first approach.

Last edited: Dec 4, 2012
7. Dec 4, 2012

Dick

If you want another approach look at the function f(x)=(1+1/x)^(x+1). Form the log and then take the derivative and try to show it's negative for x>=1.

8. Dec 5, 2012

hedipaldi

I search for a solution concerning sequences only without using functions.

9. Dec 5, 2012

micromass

Most results using sequences can be proven most quickly using functions. Is there a special reason that you don't want to use functions??

Anyway, you need to prove

$$\left(1+\frac{1}{n}\right)^{n+1}\geq \left(1+\frac{1}{n+1}\right)^{n+2}$$

Can you algebraically manipulate the above to something suitable?

10. Dec 5, 2012

hedipaldi

I tried the binom expansion but it was not efficient.

11. Dec 5, 2012

micromass

Try something else then. Try to make $1+\frac{1}{n}$ and the other one into one fraction. Then rearrange stuff.

12. Dec 6, 2012

hedipaldi

Up to now nothing works here.

13. Dec 6, 2012

micromass

There are already 12 posts in this thread. I have seen many good suggestions here. But I have yet to see an attempt of you. If you say that "nothing works", then what did you try?? Can you show us exactly what you tried?? Why doesn't it work.

If you expect to be spoonfed, then this thread will be locked.

14. Dec 6, 2012

hedipaldi

Try to solve and see the problem for yourself.This is not so easy as the increasing sequence in answer #9.

15. Dec 6, 2012

micromass

I did solve it myself.
So please make an attempt or this will be locked. If you say that nothing works, then surely there must be something that you tried?

16. Dec 6, 2012

hedipaldi

I need a solution without using functions and derivatives.I tried the binom expansion,the Bernouli inequality and more.the binom expansion may by efficient but i suspect that there must be a simpler way.

17. Dec 6, 2012

micromass

Why did you ignore my post 11?
There is a very simple way. You need to prove the inequality in post 9. First put
$$1+\frac{1}{n}~~\text{and}~~1+\frac{1}{n+1}$$
into one fraction. Then rearrange the inequality in post 9. Play around with it. What do you get?

18. Dec 6, 2012

solved.