This chapter turned every question about a series into a question about a sequence, then built tools for the cases that arise most. The questions here arrive shuffled, and deciding which tool a question calls for is the part that carries over.

The bench now holds the comparison, the sign condition, the two tests and the warning about order. Reaching for the right one is the skill; each tool on its own was the easy part.

A series is the limit of its partial sums. Non-negative terms converge exactly when those sums are bounded. Absolute convergence gives convergence and makes reordering safe, while conditional convergence gives neither. Two tests compare a series with a geometric one.

• Vanishing terms are necessary for convergence and never sufficient.
• Comparison and both tests require the terms to be handled through their absolute values.
• Absolute convergence is what makes a sum safe to reorder and safe to compute.
• At a ratio or root of exactly one, nothing has been learned.
• Order to try things in: term test, then geometric or $p$-series recognition, then comparison, then ratio or root, then the alternating test, and only then anything harder.
• Ask separately whether a convergent series converges absolutely; the answer decides whether it may be rearranged.

Great work! You now know how to:

• Choose the right test first and not third
• Sum a geometric series and recognise a p-series
• Decide absolute against conditional convergence, and say what follows