PLEASE help with 2 tricky series problems!

1) Verify that the series:

1-(1/2)+(1/2)-(1/3)+(1/2)-(1/3)-(1/4)+(1/3)-(1/4)+(1/3)-(1/4)+...

diverges and explain how this does not violate the basic theorem on alternating series.

[To my knowledge, the basic theorem they are referring to is that the summation from 0 to infinity of (-1)^k ak converges IFF, if and only if, ak-->0.]

2) Let a0, a1, a2,... be a nonincreasing sequence of positive numbers that converges to 0. Does the alternating series SUMMATION [(-1)^k]ak necessarily converge?

Thanks in advance for the help!