10.9 Determining Absolute or Conditional Convergence

A series is absolutely convergent if the series of absolute values ${\sum |a_n|}$ converges. It is conditionally convergent if the original series ∑ a_n converges, but ${\sum |a_n|}$ diverges. If the original series itself does not converge, then it is simply divergent. This distinction matters because absolute convergence is stronger: if a series converges absolutely, then it definitely converges. In AP Calculus BC, you usually determine this by first testing ∑ a_n with an appropriate positive-term test and then, if needed, testing the original alternating or signed series separately.

You should think of absolute convergence as asking whether the signs really matter. If the series still converges after all signs are made positive, then it is absolutely convergent. If it only converges because the positive and negative terms cancel in a careful way, then it is conditionally convergent. This is why alternating series often lead to conditional convergence: the original series may pass the alternating series test, while the absolute-value version behaves like a divergent harmonic or p-series.