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"?