# Alternating Series Question

1. May 2, 2012

### Bipolarity

Prove that $$\sum^{∞}_{n=1}(-1)^{n}$$ diverges.

I realized that the alternating series test can only be used for convergence and not necessarily for divergence. I might have to apply a ε-δ proof (Yikes!) which I have never been good at so please help me out.

BiP

2. May 2, 2012

### micromass

Staff Emeritus
What about the sequence of terms?

3. May 2, 2012

### Bipolarity

Hmm good point. If we can prove the sequence does not converge to 0, we have proved that the series diverges. How can we do that? Shall I look at the limit definition and look for an ε that invokes a necessary contradiction?

BiP

4. May 2, 2012

### micromass

Staff Emeritus
Yep, looks like a good plan. Try to do that.

5. May 2, 2012

### Staff: Mentor

Look at the Nth term test for divergence.

6. May 2, 2012

### HallsofIvy

Staff Emeritus
Seems to me the simplest thing to do is to show that the sequence of "partial sums",
$S_n= \sum_{i= 1}^n (-1)^n$
does not converge.