I'm wondering if you could use the ratio test for this:

|an+1/an|, So you would find the an+1 term:

[(-1)^(k+1)]*(2^(k+1))/ln(k+1)

Now, do the ratio:

[[(-1)^(k+1)]*(2^(k+1))/ln(k+1)] / [((-1)^k)*(2^k)/ln(k)]

Can be rewritten as:

[[(-1)^(k)*(-1)]*(2^(k)*2))*ln(k)] / [ln(k+1)((-1)^k)*(2^k)]

Now, you can cancel the (-1)^k term and the 2^k term from the numerator and the denominator to get:

(-1)*2*ln(k) / (ln(k+1))

Remember, you take the absolute value of the ratio so it reduces to:

2lnk/ln(k+1)

Now I don't know how to show that as k --> infinity, ln(k) and ln(k+1) are essentially equal. Then, you would be left with a ratio of 2. Since 2>1, the series would be said to diverge. I don't really know if this helps or not.