Convergence of Alternating Series: True or False? Explanation and Examples

[Spruance]

True or false, justify your answer

1) If $$\sum_{n=1}^\infty a_n$$ converges, then $$\sum_{n=1}^\infty (-1)^n \ a_n$$ converges.

2) If $$\sum_{n=1}^\infty a_n$$ converges and $$\sum_{n=1}^\infty (-1)^n \ a_n$$ converges, then $$\sum_{n=1}^\infty a_n$$ converges absolutely.

3) If $$\sum_{n=1}^\infty a_n$$ converges absolutely, then $$\sum_{n=1}^\infty (-1)^n \ a_n$$ converges absolutely.

Can anybody help me with this one?

Thanks in advance

 

[Hootenanny]

your [t e x] tags must be in lower case.

~H

 

[Spruance]

Thank you, Hootenanny. That fixed everything!

 

[Hootenanny]

RE: Your question, what are the criteria for absolute convergence?

~H

 

[HallsofIvy]

Or, what is the definition of "absolute convergence"?