can the sum be zero since they are the sums of positive terms and therefore
the sum must be bigger than the first term.
The first is the harmonic series, which can be shown to diverge by grouping
Each group after the first has terms and the smallest term is and so each is greater so:
which shows that the series is divergent.
Showing that the second converges is fairly simple, the integral test should do
the job. Determing what it sums to is more tricky but Robin Chapman give a
number of methods here.