# Limit of sequence lim n (1-cos(2/n))

1. Feb 21, 2013

### izen

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

2. Relevant equations

3. The attempt at a solution

above picture !!

2. Feb 21, 2013

### jbunniii

You can write
\begin{align} 1 - \cos\left(\frac{2}{n}\right) = 2\left[\frac{1}{2} - \frac{1}{2}\cos\left(\frac{2}{n}\right)\right] &= 2\sin^2\left(\frac{1}{n}\right) \\ \end{align}

3. Feb 21, 2013

### LCKurtz

$$\frac 1 {\frac 1 n}$$doesn't go to 0 as $n\to\infty$.

Another method is to use L'Hospital's rule on$$\frac{1 - \cos{\frac 1 n}}{\frac 1 n}$$

4. Feb 21, 2013

### izen

where is 'n' gone?

5. Feb 21, 2013

### izen

thanks LCKurtz

6. Feb 21, 2013

### jbunniii

It didn't go anywhere. I was just suggesting how to rewrite the hard part. Now if we substitute back, we get
$$n\left[1 - \cos\left(\frac{2}{n}\right)\right] = 2n \sin^2\left(\frac{1}{n}\right)$$
Can you see what to do now?

7. Feb 21, 2013

### izen

Thanks jbunniii