The basic question we wish to answer about a series is whether or not the series converges. If a series has both positive and negative terms, we can refine this question and ask whether or not the series converges when all terms are replaced by their absolute values. This is the distinction between absolute and conditional convergence, which we explore in this section.
This example shows that it is interesting to consider the role that negative terms play in the convergence of a series.
- A series converges absolutely if converges.
- A series converges conditionally if converges but diverges.
We then refine the basic question about a series (“does the series converge or diverge?”) to the following, more subtle, question: “does the series converge absolutely, converge conditionally, or diverge?”
By definition, a series converges conditionally when converges but diverges. Conversely, one could ask whether it is possible for to converge while diverges. The following theorem shows that this is not possible.
Said differently, if a series converges, then the series must also converge. It is not hard to see why this is true. The terms of any sequence (possibly containing negative terms) satisfy the inequalities If we assume that converges, then must also converge by the Comparison Test. But then the series converges as well, as it is the difference of a pair of convergent series:
The series contains both positive and negative terms, but it is not alternating. This makes it difficult to apply our standard tests to determine whether the series converges directly. On the other hand, consider the series By design, all of its terms are nonnegative. Moreover, since , we have the comparison It follows by the Comparison Test that converges. We conclude that converges absolutely, and the Absolute Convergence Theorem implies that it must therefore converge.