Absolute and Conditional Convergence

Absolute convergence simply means a series still adds up to a finite number even after you make every term positive. The name comes from absolute value: you're testing $|a_n|$, the terms with their signs removed. Conditional convergence means the series only converges on the condition that you keep its signs (and, as you'll see, its order) exactly as given. This is good to know because absolutely convergent series are safe to handle like ordinary sums, while conditionally convergent ones can play tricks on you. The statement $\sum |a_n|$ converges $\implies \sum a_n$ converges is just two pieces: $\sum |a_n|$ (add up the sizes, ignoring signs) and $\sum a_n$ (the real series, signs and all). It looks fussy, but it's really one sentence, so bear with it.

Last lesson ended with a strange fact: $\sum \frac1n$ diverges, but flip every other sign and it converges. So signs can rescue a series. That raises a natural question: when a series with mixed signs converges, is it converging because the terms are small, or only because they cancel? Those turn out to be very different situations.

Definitions.

• Absolutely convergent: $\sum |a_n|$ converges.
• Conditionally convergent: $\sum a_n$ converges but $\sum |a_n|$ diverges.

Every series lands in exactly one bucket: absolutely convergent, conditionally convergent, or divergent.

Why absolute convergence implies convergence. It makes sense that it should: if the sizes alone add up to something finite, putting in minus signs only adds cancellation, and cancellation can't make things worse. To prove it, notice $0 \le a_n + |a_n| \le 2|a_n|$, since $a_n + |a_n|$ is either $0$ or $2|a_n|$. So $\sum (a_n + |a_n|)$ converges by comparison, and subtracting the convergent $\sum |a_n|$ leaves $\sum a_n$.

Rearrangements. With a finite sum, order never matters: $1 + 2 + 3$ is $3 + 1 + 2$. An absolutely convergent series keeps that property, so you can reorder it any way you like without changing its sum. A conditionally convergent one can't. Think about why. Its positive terms add up to $+\infty$ on their own, and so do its negative terms. The finite sum only comes from interleaving them in a careful balance, and if you change the balance, you change the answer. Riemann proved it can be rearranged to add up to any number you choose, or to diverge. For example, taking one positive term of the alternating harmonic series followed by two negative ones,

half the original sum, with exactly the same terms. There's an animation further down that shows a different rearrangement landing on $1.5\ln 2$.

A common mistake. It's tempting to think "converges" settles everything, so the absolute/conditional label is just bookkeeping. It isn't. The label tells you whether you're allowed to treat the series like a normal sum. Reordering, grouping differently, or splitting it into positive and negative parts are all safe for absolute convergence and can all go wrong for conditional convergence.

Strategy. Test $\sum |a_n|$ first with the positive-term tests you already know. If it converges, you're done, and you've got the stronger result. If it diverges, try the alternating series test on $\sum a_n$.

Numerical code adds numbers in whatever order is fastest, often in parallel. That's only safe for absolutely convergent sums; a conditionally convergent sum can give different answers depending on the order.