A convergent series \(\sum a_{n}\) is said to be conditionally convergent if the absolute value series \(\sum |a_{n}|\) diverges (in other words, if the original series is not absolutely convergent). Many alternating series are conditionally convergent, such as the alternating harmonic series \(\frac11 - \frac12 + \frac13 - \frac14 \pm \dots\). The Riemann series theorem is the counter-intuitive fact that we can rearrange the terms of a conditionally convergent series to get it to sum to any real number, or \(\pm\infty\).