Series as accumulated sequences

In brief

Cut a cake in half, eat one half, and set the other half aside. Cut what remains in half again, eat that, and set the remainder aside. Repeat forever. At every step you have eaten one half, then a quarter, then an eighth, an endless list of ever-smaller pieces, and yet no matter how long you continue, you can never have eaten more than the whole cake. This is not a trick: the infinite list of pieces one half, one quarter, one eighth, and so on genuinely adds up to exactly one whole cake, no more and no less. A series is the name for exactly this kind of object, an infinite list of terms considered together with its running total, and the central question this entry answers is when such a running total, extended forever, settles on a single finite number rather than growing without bound.

The full treatment

First look: halving a cake forever

Track how much cake has been eaten after each cut. After the first cut, one half. After the second, one half plus one quarter, which is three quarters. After the third, seven eighths. After the fourth, fifteen sixteenths. Each running total is closer to one than the last, and the gap remaining, one half, then one quarter, then one eighth, shrinks by half at every step. No finite number of cuts ever reaches the whole cake exactly, but the running total is squeezed arbitrarily close to one, in exactly the sense of a convergent sequence. The cake was never divided by adding infinitely many numbers all at once; it was divided by a sequence of ordinary finite running totals that itself converges.

Defining a series as a sequence of partial sums

Given an infinite list of terms a_1, a_2, a_3, and so on, define the sequence of partial sums: S_1 = a_1, S_2 = a_1 + a_2, S_3 = a_1 + a_2 + a_3, and in general S_N is the sum of the first N terms. The series, written as the sum from n equals 1 to infinity of a_n, is said to converge to a value S exactly when the sequence S_1, S_2, S_3, and so on converges to S in the ordinary sense already established for sequences: for every tolerance, all but finitely many of the partial sums land within that tolerance of S. This definition is doing real work: it replaces the suspicious phrase "add up infinitely many numbers" with the well-understood question of whether an ordinary, term-by-term sequence of finite sums converges.

Deriving the geometric series from a telescoping trick

Take the running total of a geometric series, where each term is the previous one times a fixed ratio r: S_N = a + ar + ar^2 + up to ar^(N-1). Multiply this whole sum by r: rS_N = ar + ar^2 + up to ar^N. Subtract the second expression from the first. Every middle term cancels, leaving S_N minus rS_N equals a minus ar^N, or S_N times (1 minus r) equals a times (1 minus r^N). Dividing through, S_N = a(1 minus r^N) divided by (1 minus r), valid whenever r is not equal to 1. Now take the limit as N grows without bound. If the size of r is less than 1, then r raised to higher and higher powers shrinks to zero, so S_N converges to a divided by (1 minus r). This is exactly the cake example, with a equal to one half and r equal to one half, giving a limit of (1/2) divided by (1 minus 1/2), which equals one, matching the whole cake exactly. If r is 1 or larger in size, r^N does not shrink to zero, and the series does not converge to a finite value.

The harmonic series: a warning that shrinking terms are not enough

It seems reasonable to guess that a series converges whenever its individual terms shrink toward zero, since the cake example's terms did exactly that. The harmonic series, the sum of one plus one half plus one third plus one quarter and so on forever, is the classic proof that this guess is false. Group the terms in blocks: one third plus one quarter is greater than one quarter plus one quarter, which is one half. One fifth plus one sixth plus one seventh plus one eighth is greater than four times one eighth, which is again one half. This grouping trick, due to Nicole Oresme in the fourteenth century, can be repeated forever, and each new block adds at least another one half to the running total. Since you can always find another one half's worth of terms further along, the running total is not bounded above by any fixed number, and the sequence of partial sums does not converge. The individual terms of the harmonic series shrink to zero, yet the series still diverges. Shrinking terms are necessary for convergence, since a series whose terms do not shrink to zero certainly cannot converge, but shrinking terms alone are never sufficient.

What more is needed to decide convergence

Because term-shrinking is not enough, deciding whether a given series converges is a genuine, case-by-case question, and mathematics has developed several tools for it beyond the geometric formula above. One common strategy is comparison: if every term of an unfamiliar series is smaller than the corresponding term of a series already known to converge, the unfamiliar series converges too; if every term is larger than the corresponding term of a series known to diverge, the unfamiliar series diverges as well. These comparison tools do not remove the need to check each new series individually, but they let that checking build on cases already settled rather than starting from scratch every time.

Lineage

Zeno's ancient paradoxes of motion already implicitly wrestled with the question of whether an infinite sequence of ever-smaller distances could add to a finite total, without a rigorous answer. Nicole Oresme's fourteenth-century grouping argument that the harmonic series diverges is one of the earliest fully rigorous results in the entire subject, remarkable for predating both algebraic notation and calculus by several centuries. Through the seventeenth and eighteenth centuries, Newton, Leibniz, and especially Leonhard Euler manipulated series extensively, often with dazzling results but without a settled definition of what it meant for an infinite sum to have a value. Augustin-Louis Cauchy's 1821 Cours d'Analyse supplied that missing definition, tying a series' sum rigorously to the convergence of its sequence of partial sums, the same framework used throughout this entry.

The strongest case for it

Treating a series as a convergent sequence of partial sums makes the entire subject of infinite sums checkable rather than a matter of formal manipulation that might silently produce nonsense. It is the machinery underneath every later use of infinite sums in mathematics and science, from representing a periodic signal as a sum of waves to representing a function as an infinite polynomial, since every one of those constructions needs to know, in the same precise sense developed here, whether its infinite sum actually reaches a definite value.

The strongest case against it

Convergence for series is more fragile than it looks. A series can converge only because of cancellation between positive and negative terms, called conditional convergence, and such a series can be made to add up to any value at all, or to diverge entirely, simply by reordering its terms in a different sequence, a genuinely surprising result due to Bernhard Riemann. This shows that the ordinary intuition that addition does not care about order, true for any finite sum, fails outright for some infinite sums. Two misconceptions deserve direct naming. First, as the harmonic series shows, terms shrinking to zero never guarantees convergence by itself. Second, a convergent series cannot always be freely rearranged without changing its value; that freedom is only guaranteed when the series converges absolutely, meaning the series of the sizes of its terms, ignoring sign, also converges.

Where it stands now

The definition of a series' sum as the limit of its partial sums has been settled since Cauchy and is not in dispute. What remains a live, worked question, exactly as with sequences, is whether any particular series of interest converges and how quickly, which is precisely the practical concern underneath Taylor approximation and every other use of infinite series to represent otherwise intractable quantities.

Test yourself

A financial contract promises a payment of one hundred dollars at the end of this year, then half that amount the following year, then half of that the year after, continuing by halving forever. Using the geometric series result derived above, find the total value of all the payments added together, and state clearly why this total is finite despite there being infinitely many payments. Then construct a second, different payment schedule whose individual payments also shrink toward zero every year but whose total value is unbounded, modeled on the harmonic series argument, and explain exactly which step of your construction guarantees the payments never stop accumulating into an ever-larger sum.