A series is convergent if the sequence of its partial sums converges to a finite value. If the sequence of partial sums does not converge to a finite value the series is divergent. For example, the series 11+11+11+... (Grandi's series) is divergent because the sequence of partial sums alternates between 1 and 0. The series 1+1/2+1/3+1/4+... (the harmonic series) is divergent because the sequence of partial sums grows without bound.
Note that your series, 11/2+1/31/4+... differs from the harmonic series only in the signs of the terms. Your series is called the alternating harmonic series. There's a fairly simple test for convergence for alternating series (series whose elements alternate between positive and negative). Such a series is convergent if the sequence comprising the absolute values of the terms the series is monotonically decreasing and converges to zero as n approaches infinity. Your series passes this test and hence will converge to some finite value.
Compare your series to 11/2+1/41/8+... While the tricks that you tried to use on your series won't work on that series, they will work on this one. The reason is that the corresponding series 1+1/2+1/4+1/8+... is also convergent. Series such as 11/2+1/41/8+... are called absolutely convergent series. Rearrange the terms of an absolutely convergent series and you always get the same sum. Different arrangements of the terms of a conditionally convergent series (one that is not absolutely convergent) can yield different sums. In fact, you can rearrange the terms of a conditionally convergent series to give any sum you want.
