A series with mixed signs can converge for two quite different reasons: because its terms are genuinely small, or because its positive and negative parts happen to cancel. Taking absolute values before summing tells the two apart. A series that still converges once the signs are removed is called absolutely convergent, and it is far better behaved than one that is not.

Two accounts both end the year at zero. One had almost no transactions; the other had enormous sums moving in and out that happened to balance. The totals agree, and the accounts are not remotely alike. Summing the sizes of the transactions is what tells them apart.

The absolute series has non-negative terms, so the previous lesson applies to it in full. That is where all the strength comes from: questions about a signed series are routed through one that has no signs.

• $\sum a_k$ converges absolutely when $\sum \lvert a_k \rvert$ converges.
• It converges conditionally when it converges but $$ does not.

Split each term into the part that is positive and the part that is negative, both taken as non-negative numbers. Each is dominated by the absolute value, so each series converges by comparison, and the original is their difference.

Separate the rise from the fall, keeping both as non-negative numbers.

Neither part can exceed the size of the term it came from.

Both are non-negative series under a convergent one.

Exactly one of the two parts is non-zero, and it carries the right sign.

The difference of two convergent sequences of partial sums converges.

$\sum \tfrac{(-1)^{k+1}}{k^2}$ converges absolutely, since converges.

Removing the signs leaves a convergent series, so the original is absolutely convergent.

$\sum \tfrac{(-1)^{k+1}}{k}$ converges, while $\sum$ does not.

The cancellation is doing all the work, which is what conditional means.

One counterexample settles the converse, and the alternating harmonic series is it.

Both parts are non-negative, so comparison applies to each of them separately.

Ask first whether the sizes are summable. The answer sorts every convergent series into two kinds that behave very differently.

• Absolute convergence means the sizes are summable, and it implies convergence.
• Conditional convergence is convergence without that, and the alternating harmonic series is the standard example.
• For non-negative terms the two notions coincide.
• Every test for non-negative terms tests absolute convergence when applied to the sizes.

Great work! You now know how to:

• Sort a convergent series into absolute or conditional
• Apply a non-negative test to the sizes
• Say which implication holds and give the counterexample to the other