# Limit of a sine

1. Jul 8, 2008

### dirk_mec1

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

Find this limit

$$\lim_{n \rightarrow \infty} n\cdot sin(2 \pi n!e)$$

2. Relevant equations
hint: use the euler power series

3. The attempt at a solution
Well I know that:

$$e = \sum_{k=0}^{\infty} \frac{1}{k!}$$

But how can I use this information?

2. Jul 8, 2008

### G01

That is the power series representation for e.

What is the power series representation of the sine function?

3. Jul 8, 2008

### dirk_mec1

$$\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}+\cdots$$

Do I have to put one series into the other?

4. Jul 8, 2008

### m_s_a

Unfortunately, I have erred in solution
Equal second incorrect

5. Jul 8, 2008

### dirk_mec1

What do you mean?

6. Jul 8, 2008

### Dick

Look at the terms in the argument of the sine for a large value of n after you've used the power series expansion for e. Many are integer multiples of 2pi. Now look at the first one that isn't. Can you justify ignoring the terms after that? Think l'Hopital.

7. Jul 8, 2008

### G01

Good call Dick. I don't know what I was trying to do when I posted before about the sine expansion. That's what I get for posting at work...

8. Jul 9, 2008

### vipulsilwal

n!*e comes out to be nP0+nP1+nP2+nP3.................all integers
sin(2pi*integer)==0
infinity * 0=0

9. Jul 9, 2008

### dirk_mec1

You're wrong the answer isn't zero.

10. Jul 9, 2008

### vipulsilwal

would you please tell where i am going wrong....

11. Jul 9, 2008

### Dick

Put in the infinite series for e. A lot of terms are multiples of two pi. They aren't all. The first one that isn't contributes 2*pi/(n+1) to the sine argument. (There are more but they can be ignored relative to this one for large n).

12. Jul 9, 2008

### vipulsilwal

how n+1??
both the n(one of the n! and other of the e expansion) go upto infinity..

so it would expand to:
2pi + 2pi(n) + 2pi(n)(n-1)+ 2pi(n)(n-1)(n-2) +................................2pi(n!)
2pi(n!) being the largest one.

13. Jul 9, 2008

### Dick

Insert the power series first. Then let n go to infinity. Don't try and do both at the same time. The 2*pi*n! term comes from the k=0 term in the expansion, 2*pi comes from the k=n term, right? What about the k=n+1 term?

14. Jul 9, 2008

### vipulsilwal

i am not getting it.
could you please send me the whole solution..

15. Jul 9, 2008

### Dick

We aren't in the business of providing 'whole solutions'. If you have another question ask it. I don't see what you aren't 'getting'. Put infinite series in sine argument - decide what terms are important. Take limit.