Alternating harmonic series
= Alternating harmonic series
The alternating harmonic series $\sum_{n\geq1}(-1)^{n+1}/n$ converges. Its even partial sums increase, its odd partial sums decrease, and the two limits agree because their difference is the next term. In particular, every even partial sum is a lower bound and every odd partial sum is an upper bound for the limit.
This alternates the signs of the terms of the <harmonic series>, which itself diverges.