Does the Series Sum of (-2)^n / 3^(n+1) from 0 to Infinity Diverge?

[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 +?

 

[P3X-018]

Are you sure you wrote it correct? Because the series

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

does converge...

 

[frasifrasi]

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

 

[Dick]

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

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

 

[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)

 

[Dick]

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