
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.

Recall that the basic question about a series that we seek to answer is “does the series converge?” It turns out that if the series contains negative terms, there is an interesting refinement of this question. This is illustrated by the following example.

This example shows that it is interesting to consider the role that negative terms play in the convergence of a series.

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?”

Does the series converge absolutely, converge conditionally, or diverge?
The series converges conditionally. The series converges absolutely. The series diverges.
Does the series converge absolutely, converge conditionally, or diverge?
The series converges conditionally. The series converges absolutely. The series diverges.

By definition, a series converges conditionally when $\sum a_n$ converges but $\sum |a_n|$ diverges. Conversely, one could ask whether it is possible for $\sum |a_n|$ to converge while $\sum a_n$ diverges. The following theorem shows that this is not possible.

Said differently, if a series $\sum |a_n|$ converges, then the series $\sum a_n$ must also converge. It is not hard to see why this is true. The terms of any sequence $\{a_n\}$ (possibly containing negative terms) satisfy the inequalities If we assume that $\sum |a_n|$ converges, then $\sum (a_n + |a_n|)$ must also converge by the Comparison Test. But then the series $\sum a_n$ converges as well, as it is the difference of a pair of convergent series: