alternating harmonic series
First, notice that the series is not absolutely convergent. By taking the absolute value of each term, we get the harmonic series, which is divergent. There are several ways to show this, and we invite the reader to the entry on harmonic series for further exploration.
Next, to show that the series (1) converges, we use the alternating series test (http://planetmath.org/AlternatingSeriesTest): since
the alternating series (1) converges.
Other examples of conditionally convergent series can be discovered using variants of the alternating harmonic series. For instance, the following series
can easily be shown to be conditionally convergent. Here is another example, more of a generalization, called the :
(2) convergence absolutely convergent conditionally convergent divergent