# Lightning SERIES question

1. Nov 4, 2007

### frasifrasi

can anyone explain why the summation from 0 to infinity of

(-2)^(n)/3^(n+1) diverges?

- Is it simply because the terms bounce between - and +?

2. Nov 4, 2007

### P3X-018

Are you sure you wrote it correct? Because the series

$$\sum_{n=0}^{\infty} \frac{(-2)^n}{3^{n+1}}$$

does converge...

3. Nov 4, 2007

### frasifrasi

I have in the answer that it diverges...could you explain how you arrived at that?

4. Nov 4, 2007

### Dick

It's (1/3)*(-2/3)^n. It's a geometric series. You can even sum it.

5. Nov 4, 2007

### frasifrasi

I see the light! I guess the answer key was wrong. But hey,

What if had the SEQUENCE (-2)^(n)/3^(n+1) , how could I show that it converges to 0?

could I also "simplify" it to (1/3)*(-2/3)^n and say that since r > -1, it converges to 0?

(by the fact that for a sequence r^n , the sequence converges for -1 < r <= 1)

6. Nov 4, 2007

### Dick

If you mean |r|<1, then yes, the sequence converges to zero.