If this series is convergent it is conditionaly convergent. But a conditionaly convergent series can be rearranged to sum to anything, so you cannot rearrange the terms.

Now the sum of pairs of terms gives the harmonic series, so the partial sum of 2n terms is equal to the partial sum of n terms of the harmonic series, so the series cannot converge (since if it did every subsequence of the sequence of partial sums would have to have the same finite limit).

RonL