# Prove sum identities

1. May 1, 2007

### DieCommie

1. The problem statement, all variables and given/known data
Prove the following identities:
$$\sum_{n=0}^{\infty} r^n \cos(n\theta) = \frac{1-r\cos(\theta)}{1-2r\cos(\theta)+r^2}$$

$$\sum_{n=0}^{\infty} r^n \sin(n\theta) = \frac{r\sin(\theta)}{1-2r\cos(\theta)+r^2}$$

2. Relevant equations
$$\cos(n\theta) = \frac{1}{2}(e^(in\theta)+e^(-in\theta))$$

3. The attempt at a solution
I get it to this point...

$$\frac{1}{2}( \sum_{n=0}^{\infty}(re^(i\theta))^n+\sum_{n=0}^{\infty}(re^-(i\theta))^n$$

But I dont know what to do next! r could be less than or greater than one. I need to do the general case where r could be either....

Also, Im not sure that the 'relevent equation' is really relevent

Any help/tips would be greatly appreciated, Thx!

Last edited: May 1, 2007
2. May 1, 2007

### Dick

You are now looking at a pair of geometric series. The next step should be easy. I wouldn't worry about the case r>1. In general it should diverge except for special angles.

3. May 3, 2007

### MaGG

Wouldn't you just use the following rule for an infinite geometric series:

$$\sum\limits_{k = 1}^\infty {ar^{k} = \frac{a}{{1 - r}}}$$

Last edited: May 3, 2007
4. May 3, 2007

### DieCommie

Yes, but only if R<1. The problem conspicuously leaves out that stipulation, which is where my confusion set it.

I went ahead and solved it assuming R<1 and ignoring the other case.... I hope thats good enough for some credit.

5. May 3, 2007

### MaGG

Did you take $$r=re^{i\theta}$$ and $$a=1$$? Now you've gotten me wanting to solve this problem, lol...

6. May 3, 2007

### DieCommie

I took $$re^{i\theta} = r(cos\theta + isin\theta)$$ and a=1. I have it completed assuming r<1. We will look at later.

7. May 4, 2007

### Dick

If r=1 then the series is going to be, at best, only conditionally convergent (r>1, not even that except for some silly special cases like theta=pi). I would really only worry about r inside of the radius of convergence of the series. You can't 'prove the identities' if the sum of the series doesn't exist.

Last edited: May 4, 2007