Absolute versus conditional convergence

Understand the idea

Absolute convergence means the series of term magnitudes converges. It guarantees convergence of the original signed series. Conditional convergence means the original series converges but its absolute-value series diverges. Both parts of that definition need justification.

Start by testing the magnitudes when a familiar positive series appears. A convergent p-series or a ratio-test result below one can establish absolute convergence immediately. If the magnitude series diverges, the original signed series still needs its own analysis.

An alternating-series argument can establish convergence when absolute convergence fails. That combination gives conditional convergence. If the original terms do not approach zero, the series is divergent and is neither absolutely nor conditionally convergent.

\sum|a_n|<\infty\Rightarrow\sum a_n\text{ converges};\qquad \text{conditional}=\text{convergent but not absolutely convergent}

A method you can apply

Study $\displaystyle \sum|a_n|$ with an appropriate positive-series test.
If it diverges, investigate whether the original signed series converges.
State both conclusions and use the classification that matches them.

Worked example

Classify $\displaystyle \sum_{n=1}^{\infty}(-1)^{n-1}/\sqrt n$ .

The magnitudes $\displaystyle 1/\sqrt n$ decrease to zero, so the alternating-series test establishes convergence.
The absolute-value series is $\displaystyle \sum n^{-1/2}$ .
This is a divergent p-series with $\displaystyle p=1/2$ .

The original series converges conditionally.

Try it yourself

Classify $\displaystyle \sum_{n=1}^{\infty}(-1)^n/n^2$ .

Show solution and reasoning

The absolute-value series is $\displaystyle \sum1/n^2$ , which converges because $\displaystyle p=2>1$ . Therefore the signed series converges absolutely; it should not be called conditionally convergent.