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!
If I have interpreted your intention correctly:
After the third term the terms may be grouped into threes so:
But
So:
So the series must diverge as the partial sums are greater than a constant plus the partial sums of a multiple of the harmonic series which diverges.
RonL