The sum that fits in one square

To cross a room, first cross half of it, then half of what remains, then half of that. The crossing is cut into infinitely many stages, each of which takes some time, and Zeno’s question is how infinitely many positive quantities can add up to a finite one.

The answer is a square.

The pieces are 1/2, 1/4, 1/8 and so on. They sit inside one square, they do not overlap, and there is no gap. The figure asserts all three: it checks that the pieces together with the remaining corner tile the square exactly, with nothing outside it, no overlap and no gap.

What the picture settles and what it does not #

The square settles a specific question: can the partial sums exceed one? They cannot, because they are areas of disjoint pieces of a square of area one. That is a complete argument and it needs no limits.

It does not by itself settle that the sum is one. Every partial sum is less than one, and a collection of quantities all less than one could perfectly well be heading for three quarters. What the picture adds is the remainder: after k cuts, what is left uncut is exactly 2⁻ᵏ of the square, and that shrinks below any positive number that anyone names.

So the sum is one, in the only sense the phrase has: the partial sums come and stay arbitrarily close to it. That is a definition rather than a discovery, and it is worth being explicit that a definition is what is doing the work. Nothing infinite is being added up. A limit of finite additions is being taken, and the limit is given the name of a sum.

Zeno’s question, and which half of it this answers #

The paradox is usually stated about distance and it is not really about distance. Cutting a journey into halves is a description of the journey and describing something in infinitely many pieces does not make it longer.

The version with teeth is about time. Each stage takes some time; there are infinitely many stages; how can the total time be finite? And that version is answered by exactly the picture above, provided the stages’ durations also halve — which they do at constant speed, since each stage covers half the distance of the last.

So the resolution is not that the stages are illusory. They are real, there are infinitely many of them, and their durations add to a finite total because they shrink geometrically. What Zeno’s argument lacks is the observation that infinitely many positive numbers can have a finite sum, and that observation took two thousand years to be made precisely rather than merely asserted.

The other half of the question is not answered here and is worth flagging. Whether physical time is infinitely divisible is not a mathematical matter, and nothing on this page bears on it. What the mathematics settles is that the infinite divisibility is not by itself a contradiction.

Archimedes, doing this without the notation #

The earliest careful version of this argument is Archimedes’ calculation of the area inside a parabola, and it is worth recounting because of how it handles the limit.

He inscribes a triangle in the parabolic segment, then two smaller triangles in the gaps, then four smaller ones, and shows that each generation has exactly a quarter of the area of the one before. The total is 1 + 1/4 + 1/16 + …, which is 4/3 of the first triangle.

What he does not do is take a limit, because the notion was unavailable. Instead he proves the area cannot be more than 4/3 and cannot be less, each by showing that a supposed difference would eventually be exceeded by the pieces already accounted for. That is the method of exhaustion, it is logically airtight, and it is enormously more laborious than the modern statement — which is a limit, defined once, and then used everywhere.

The same manoeuvre underlies adding up rectangles and every area computed by cutting into pieces. The geometric series is the case where the pieces have a fixed ratio, which is what makes the total nameable in closed form rather than merely bounded.

The same series in three disguises #

Starting the series at 1 rather than at a half gives the more familiar form, and the sum is 2. In general, the terms 1, r, r², … add to 1/(1 − r), and the amount missing after k terms is exactly rᵏ/(1 − r).

That last expression is the whole of the theory. It is exact, not an estimate, and the figure asserts it: the difference between the limit and the partial sum is computed both ways and compared.

The derivation takes one line and it is worth having, because it is completely finite. Let S be the sum of the first k terms. Then rS is the same sum with the first term dropped and one extra term on the end, so S − rS = 1 − rᵏ, which gives S immediately. No limits, no infinities, just a subtraction — and the limit is then taken in the one place where it is needed, on the term rᵏ.

Why an infinite sum needs a definition at all #

The step that deserves suspicion is the one where “the sum of infinitely many terms” is declared to mean “the limit of the finite partial sums”. That is a decision, and it is worth asking what would go wrong under a different one.

The decision is forced by wanting sums to behave like sums. Adding a constant to every term, multiplying the whole thing by a number, adding two series term by term — all of these work as expected under the partial-sum definition, and the reason is that each of them works for every finite stage and limits respect them.

Try instead to define the sum by some average of the partial sums, and a great deal survives; some divergent series even acquire values. Try to define it by something that ignores the order of the terms, and the whole enterprise runs into the rearrangement problem immediately.

What the square adds to this is a reason to be comfortable with the choice in this particular case. Under any reasonable definition, a collection of disjoint pieces of a square with a shrinking remainder ought to total the square. The definition and the picture agree, which is exactly the situation in which a definition can be trusted — and the definition then has to be carried, unchanged, into cases where no picture is available at all.

The ratio, and how much it matters #

The convergence rate is entirely the ratio’s business. At a half, ten terms leave two thousandths missing. At nine tenths, twelve terms leave nearly three units missing out of ten. At 0.999 the series still converges, to 1000, and it takes seven thousand terms to get halfway there.

This matters because “converges” is a yes-or-no property and it is almost never the useful one. Two series can both converge and be entirely different objects to compute with. What separates them is the rate, and for a geometric series the rate is exponential in a way that makes the ratio the only number that matters.

The dissection works for any ratio strictly between nothing and one, and the picture makes plain what changes: a ratio near one leaves a large remainder for a long time. The pieces are thinner and the uncut corner is stubborn.

Where the picture stops #

At a ratio of one the terms do not shrink at all and the total grows without bound, which needs no argument. Above one it is worse. The generator refuses both — a ratio outside the open interval is rejected before anything is drawn — because a figure claiming a limit that does not exist is exactly the kind of well-formed wrong picture this site’s assertions exist to prevent.

What is more interesting is what happens just below the boundary, on the other side of it, and at it.

The harmonic series has terms that shrink to nothing and a total that grows without bound. It is not geometric — the ratio between consecutive terms is 1/2, then 2/3, then 3/4, creeping towards one — and it is the standing proof that shrinking terms are not enough.

That is the sharpest thing the square can say. A dissection of a square is available exactly when the total is finite; the harmonic series has no such picture, and the reason is not that nobody has found one.

The same series in three disguises #

The geometric series turns up so often that its appearances are worth collecting.

A doubling process run backwards is one. Anything that halves at a fixed rate — a quantity, a length, a share — has a total that is finite and computable in one step, and the ratio decides the answer. That is the arithmetic behind every calculation where a process contributes a fixed fraction of what the previous stage contributed, and the fact that the answer is 1/(1 − r) rather than something requiring a sum is the single most useful consequence on this page.

A repeating decimal is one. The number 0.3333… is 3/10 + 3/100 + 3/1000, a geometric series with ratio a tenth, and its sum is (3/10)/(9/10) = 1/3. That is why every repeating decimal is a fraction and why the conversion is a formula rather than a search. It is also the honest answer to why 0.999… is 1: the series has ratio a tenth and first term 9/10, and its sum is exactly one, in the same sense as everything else on this page.

The area under an exponential is one. Cutting the region under a decaying curve into vertical strips of equal width gives a geometric series, which is why exponential decay has a finite total — and why rectangles under a curve settle down for such a curve without any care being needed.

And it is the first power series. The sum 1 + x + x² + … equals 1/(1 − x) for |x| < 1, which is the Taylor expansion of that function about zero. The radius of convergence — the reason the expansion works inside one and not outside — is exactly the boundary this essay has been circling, and the geometric series is the case where the boundary is visible without any machinery at all.

What the picture cannot show #

Nothing here shows why the ratio must be constant. The dissection cuts off the same fraction each time, and that is what makes the pieces geometric. A dissection that cut off a different fraction at each stage would produce some other series, and whether it left a shrinking remainder would be a separate question with a separate answer — which is what makes the harmonic series a genuinely different picture rather than this one with a parameter changed.

The picture is of finitely many pieces. Seven cuts are drawn and the eighth would be invisible. What the drawing establishes is a statement about every finite stage — the pieces so far fit inside the square with a specific amount left over — and the infinite statement is obtained by watching a formula rather than by looking at a picture.

The remainder is the whole content and it is the hardest thing to draw. Everything interesting is in the corner that shrinks, and the corner shrinks out of visibility long before the argument is over. A figure drawn with thirty cuts would look identical to one with seven and would be a worse picture.

And the dissection is a choice. The same series has other dissections — a triangle cut into similar triangles, a line segment cut in successive halves — and the square was picked because its pieces stay rectangular. A reader who came away thinking the square is what the series is has taken one representation for the object, which is the standing risk of a proof by picture and the reason the closed form is stated beside it.

The number this makes available #

There is a consequence of all this that is easy to pass over and is the reason the series matters beyond its own arithmetic: it is what makes decimal notation mean anything.

A decimal expansion is an infinite sum — three tenths plus one hundredth plus four thousandths and onwards — and every such sum is dominated term by term by a geometric series with ratio a tenth. So every decimal expansion converges, whatever its digits, and writing down an infinite string of digits is writing down a number rather than a hopeful gesture.

That is the foundation the whole of how closely a fraction can approximate is built on, and it is why a real number can be specified by a rule for producing digits. Without a guarantee of convergence, an expansion would be a procedure rather than a value.

It also explains the one case where the notation is genuinely ambiguous. The strings 0.999… and 1.000… are different strings and the same number, because two different geometric series can have the same sum. Every real number with a terminating expansion has exactly two, and every other real number has exactly one — a small blemish in the notation that follows directly from the fact on this page and cannot be removed without giving something else up.

Where the ladder goes next #

The rung directly above asks what happens when the terms are not a fixed ratio apart. The comparison test says a series whose terms are eventually smaller than a convergent geometric series converges too, which makes the geometric series the yardstick against which almost everything else is measured — and the ratio test is precisely the instruction to check whether a series is approximately geometric.

The other direction is towards what a sum can be rearranged into. Every series on this page has positive terms, and a positive series has a sum that does not depend on the order of its terms. Allow negative ones and that stops being true in the most spectacular way available: the same terms in a different order can be made to add to anything at all. The square is safe from that, because a dissection has no order in it — but the safety is a property of positivity rather than of the picture.