Addition does not care about order, and for a finite sum that settles the matter. For an infinite series it does not, because the value is a limit rather than a sum, and reordering the terms reorders the running totals. When the series converges absolutely the value survives any reordering. When it converges only conditionally the value can be moved anywhere at all.

Rearranging a finished pile of coins changes nothing. Rearranging the order in which they are counted changes what the running total looks like at every moment, and it is the running total, not the pile, that an infinite series is about.

A rearrangement uses exactly the same terms, each once, in a different order. The two results below are as far apart as two theorems about the same operation can be, and which one applies is decided by absolute convergence.

• An absolutely convergent series has the same sum under every rearrangement.
• Riemann's theorem: a conditionally convergent series can be rearranged to converge to any chosen number, or to diverge.
• The alternating harmonic series can therefore be made to sum to zero, to a hundred, or to nothing at all.
• A finite sum is unaffected by order, because it is not a limit.
• A rearrangement is a bijection of the index set: every term appears exactly once, and only the order changes.
• Riemann's theorem is stated here and not proved; the construction alternates between taking positive terms until the target is passed and negative ones until it is passed back, and both supplies are inexhaustible precisely because the convergence is conditional.

Any partial sum of one arrangement uses finitely many terms, all of which appear early enough in the other. So each total is bounded by the other series, and two numbers each at most the other are equal.

A rearrangement takes every term exactly once.

However far along, only finitely many have been added.

Finitely many terms cannot be spread beyond some point of the other list.

The terms are non-negative, so adding the missing ones only increases the total.

The bound holds for every partial sum, so it holds for their least upper bound.

The same argument with the two lists exchanged closes it.

The difference is that one is an act of addition and the other is a limit.

$\sum \tfrac{(-1)^{k+1}}{k^2}$ may be reordered freely.

Its absolute series converges, so every arrangement gives the same number.

Take positive terms until the total passes a hundred, then one negative, and repeat for ever.

Two numbers each at most the other are equal, which is the same move as uniqueness of a supremum.

Nothing is added, removed or repeated. Only the order in which the terms are met changes.

• A rearrangement is a bijection of the indices; nothing is added or removed.
• An absolutely convergent series has one sum under every rearrangement.
• A conditionally convergent one can be made to reach any target, or to diverge.
• A finite sum is unaffected by order, because it is not a limit.

Great work! You now know how to:

• Say what a rearrangement is and what it is not
• Name which series may be reordered safely
• State Riemann's theorem and say what makes it possible