Series (mathematics)

• 1. The Sum That Should Not End

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  A runner named Achilles chases a tortoise, and according to Zeno of Elea he can never catch it. Each time Achilles reaches where the tortoise was, the tortoise has crept forward to a new spot. Reach that spot, and the tortoise has moved again. Zeno divided the race into infinitely many sub-races, each taking a finite amount of time. He is said to have argued that continuous motion must therefore be an illusion. To the Ancient Greeks, the idea that infinitely many quantities added together could yield a finite result seemed paradoxical. Yet that very idea sits at the heart of one of the most useful tools in mathematics: the series. A series is, roughly speaking, an addition of infinitely many terms, one after another. How can an endless list of additions ever settle on a single answer? When does such a sum exist at all, and when does it spin off into nothing? Why did it take until the 17th century to tame Zeno's puzzle, and what did mathematicians discover when they finally did? The race between Achilles and the tortoise has a finite sum, and that sum is exactly the time he needs to catch up.

• 2. Catching The Limit

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  Archimedes produced the first known summation of an infinite series, using a method still used in calculus today. He applied the method of exhaustion to find the area under the arc of a parabola, and from it drew a remarkably accurate approximation of pi. The infinite additions a series describes cannot be carried out one by one in a finite amount of time. Instead, mathematicians look at the partial sums, the totals you get from adding just the first few terms. If those partial sums approach some fixed value as you take more and more of them, that value is called the sum of the series. When the limit exists, the series is called convergent, or summable; when it does not, the series is divergent. A series with only finitely many nonzero terms always converges. The mathematical side of Zeno's paradoxes was resolved through this concept of a limit, especially in the early calculus of Isaac Newton in the 17th century. The gap between the true sum and a partial sum even has a name: the truncation error, and measuring it became important enough to anchor whole fields of numerical analysis.

• 3. When Order Changes Everything

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  Grandi's series adds one, subtracts one, adds one, and so on forever, and its partial sums flip between one and zero without ever settling. Group its terms in pairs and you get a series that sums to zero. Start the pairing one term later and you get a series that sums to one. Two groupings, two different answers, from the same endless list. Finite rearrangements never change a sum, because their effects stay trapped inside a finite stretch of terms. Infinite rearrangements are another matter entirely. A series whose terms can be reshuffled to converge to a different value is called conditionally convergent. The Riemann series theorem says any conditionally convergent sum of real numbers can be rearranged to land on any real number you like, or to diverge. The alternating harmonic series is the classic case. Left alone, it sums to the natural logarithm of 2. Take the absolute value of each term and you get the harmonic series, which diverges, a divergence first proven by Nicole Oresme in the 14th century. Reshuffle the alternating harmonic series so each positive term is followed by two negatives, and the sum drops to half the natural logarithm of 2. A series that survives every rearrangement with the same answer is called unconditionally convergent, and for real and complex numbers that property is identical to absolute convergence.

• 4. An Arithmetic Of Infinities

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  Add two convergent series term by term and the result is a convergent series whose sum is the sum of the two originals. The rule has a strange edge: adding two divergent series can produce a convergent one. Pair a divergent series with the same series times negative one and every term cancels to zero. Multiply a series by a constant scalar, term by term, and a summable series stays summable. Together, addition and scalar multiplication give the set of convergent real series the structure of an infinite-dimensional vector space. Multiplication of two series is harder. The product, called the Cauchy product, does not converge as easily as a sum, but if both factors converge absolutely, so does the product, and its sum equals the product of the two sums. With multiplication added, the absolutely convergent series form a commutative ring, the same algebraic skeleton that governs ordinary numbers.

• 5. A Field Guide To Series

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  A geometric series multiplies each term by a fixed common ratio, and it converges only when that ratio is small enough in magnitude. The harmonic series, by contrast, simply adds reciprocals and diverges. A telescoping series collapses when its underlying sequence approaches a limit. The Dirichlet series, whose sum as a function gives Riemann's zeta function, converges in some regions and diverges in others. One series remains a genuine open question. Nobody knows whether the Flint Hills series converges, because its behavior depends on how well pi can be approximated by rational numbers, which itself is unknown. The largest contributions come from the denominators of the continued fraction convergents of pi, a sequence that begins 1, 3, 22, 333, 355, 103993. Each of those is close to a whole multiple of pi, which makes a certain sine value tiny and its reciprocal enormous. A puzzle this concrete, still unsolved, shows how much of the subject stays alive.

• 6. Tools For Proving Convergence

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  The simplest check is the nth-term test: if the terms do not shrink toward zero, the series diverges, though shrinking terms alone prove nothing. When every term is non-negative, a series converges exactly when its partial sums stay bounded, which turns convergence into a hunt for an upper bound. The direct comparison test pits an unknown series against a known one: dominate a convergent series and you converge, undercut a divergent one and you diverge. Comparisons against geometric series yield the ratio test and the root test, two of the most common tools. Comparisons against integrals give the integral test, while flattening a series leads to Cauchy's condensation test, the same reasoning Oresme used on the harmonic series. Conditional convergence needs its own instruments. The alternating series test, also called the Leibniz test, certifies an alternating series whose terms decrease monotonically to zero. It is a special case of the broader Dirichlet's test, and Abel's test handles related semi-convergent sums. For the alternating series, the truncation error can even be pinned down exactly, a precision prized in validated numerics and computer-assisted proof.

• 7. Series Of Functions

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  Replace numbers with functions and the terms can be added at every point of a set, giving pointwise convergence. A stronger demand is uniform convergence, where the worst error across the whole set shrinks to zero. Uniformity is prized because it preserves the good behavior of the terms. A uniformly convergent series of continuous functions has a continuous limit, and a uniformly convergent series of integrable functions can be integrated term by term. A power series is built from rising powers of a variable, and the Taylor series of a function is the most familiar example. Unless it converges only at its center, a power series converges inside a disc whose radius is called the radius of convergence, and the convergence is uniform on compact subsets of that disc. Power series have a second life as formal objects, where the plus sign is treated as a symbol of conjunction and no addition is ever performed. In that guise they drive the method of generating functions in combinatorics. Laurent series extend the idea to negative exponents and converge in an annulus rather than a disc, while Fourier series build functions from trigonometric terms.

• 8. Building The Theory

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  Mathematicians from the Kerala school in medieval India were studying infinite series around 1350 CE, deriving series expansions for trigonometric functions. Those results appeared in Sanskrit verse in Neelakanta's Tantrasangraha around 1500, then with proofs a century later in the Yuktibhasa, written in Malayalam by Jyesthadeva. In Europe, James Gregory worked on infinite series in the 17th century and published several Maclaurin series, and in 1715 Brook Taylor gave a general method for constructing Taylor series. Leonhard Euler later operated liberally with infinite series even when they did not converge, developing the theory of hypergeometric series and q-series. The push for rigor came in the 19th century. Carl Friedrich Gauss published a memoir on the hypergeometric series in 1812, and Augustin-Louis Cauchy in 1821 insisted on strict tests of convergence, beginning the discovery of effective criteria. Cauchy also showed that the product of two convergent series need not converge. Abel corrected some of Cauchy's conclusions in 1826 and stressed the role of continuity in questions of convergence. Fourier set himself the problem of expanding a function in sines and cosines, embodied in his Théorie analytique de la chaleur of 1822, though he left the question of convergence for Dirichlet to settle in a thoroughly scientific manner in 1829. The physical side of Zeno's puzzle, meanwhile, remains open, with theories of quantum gravity proposing that spacetime itself may be quantized at the Planck scale.