Can the Alternating Series Test Prove Divergence?

[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

 

[micromass]

What about the sequence of terms?

 

[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

 

[micromass]

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?

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

 

[Mark44]

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?

Look at the Nth term test for divergence.

 

[HallsofIvy]

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.