Expii

# Conditional Convergence of Series - Expii

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 $$\pm\infty$$.