Sequences and Series

Section ISequences and their limits

A sequence is an endless list of numbers a₁, a₂, a₃, …, usually given by a formula for the nth term. For a_n = 1/n the list is 1, 1/2, 1/3, 1/4, …

A sequence converges to a limit L if its terms get as close to L as you like and stay there. We write limn → ∞a_n = L. If there is no such number, the sequence diverges. The tricks from limits at infinity still work. For example

limn → ∞3n² + 1 over n² + 5 = limn → ∞3 + 1/n² over 1 + 5/n² = 3

Some sequences diverge by growing without bound, like a_n = 2ⁿ; others never settle, like (−1)ⁿ, which flips between −1 and 1. A useful fact: rⁿ goes to 0 when |r| < 1, and diverges when |r| > 1.

Section IISeries and partial sums

A series is what you get when you add the terms of a sequence: Σ_{n = 1}^∞ a_n = a₁ + a₂ + a₃ + …. You cannot do infinitely many additions, so instead look at the running totals, the partial sums

S_n = a₁ + a₂ + … + a_n

The partial sums form a new sequence. If it converges to S, the series converges and its sum is S. If not, the series diverges. Keep the two sequences apart: a_n is what you add each time, S_n is the total so far.

Section IVGeometric series

In a geometric series each term is the one before times a fixed ratio r: a + ar + ar² + ar³ + …. Multiply the partial sum by r and subtract, and almost everything cancels:

S_n − rS_n = a − arⁿ,   so   S_n = a(1 − rⁿ) over 1 − r

When |r| < 1, rⁿ goes to 0, so the series converges and

Σ_{n = 0}^∞ arⁿ = a over 1 − r,   |r| < 1

When |r| ≥ 1 (and a ≠ 0) it diverges. For 1/2 + 1/4 + 1/8 + …, a = 1/2 and r = 1/2, so the sum is (1/2)/(1/2) = 1: walk half way to a wall, then half the rest, and so on, and the total distance is exactly the distance to the wall.

Section VThe harmonic series diverges

The harmonic series is 1 + 1/2 + 1/3 + 1/4 + …. Its terms shrink to 0, so it is tempting to think the sum settles down. It does not. Group the terms in blocks of 1, 2, 4, 8, … terms:

1 + 1 over 2 + (1 over 3 + 1 over 4) + (1 over 5 + … + 1 over 8) + … ≥   1 + 1 over 2 + 1 over 2 + 1 over 2 + …

Each bracket is at least 1/2, because every term in it is at least as big as its last term: 1/3 + 1/4 ≥ 1/4 + 1/4, and the four terms 1/5 to 1/8 add to at least 4 × 1/8. So S_2^k ≥ 1 + k/2, and the partial sums pass every number. The growth is very slow: it takes 12,367 terms to pass 10.

The lesson: terms going to 0 does not guarantee that a series converges.

Section VIConvergence tests, in brief

Usually you cannot find a formula for S_n. Tests tell you whether a series converges without finding its sum. Three of the most useful:

1. The nth-term test. If a_n does not go to 0, the series diverges. (If it does go to 0, this test says nothing: think of the harmonic series.)
2. The comparison test. For positive terms with a_n ≤ b_n: if Σ b_n converges, so does Σ a_n; if Σ a_n diverges, so does Σ b_n. Good series to compare with are geometric series and the p-series Σ 1/n^p, which converges exactly when p > 1.
3. The ratio test. For positive terms, find L = limn → ∞a_n+1/a_n. If L < 1 the series converges; if L > 1 it diverges; if L = 1 the test cannot decide. It works well with powers and factorials.

Section VIIPower series and Taylor polynomials

A power series is a series with a variable in it: Σ_{n = 0}^∞ c_nxⁿ = c₀ + c₁x + c₂x² + …. It is like a polynomial that never ends. Each power series converges on an interval centred at 0 (its radius of convergence can be 0, a positive number, or infinite) and diverges outside it.

Which polynomial best copies a function f near 0? Match the value and as many derivatives at 0 as you can. That gives the Taylor polynomial of degree n about 0 (also called a Maclaurin polynomial):

P_n(x) = f(0) + f′(0)x + f″(0) over 2!x² + … + f⁽ⁿ⁾(0) over n!xⁿ

Letting n → ∞ gives the Taylor series. Four to know:

e^x = 1 + x + x² over 2! + x³ over 3! + …   (all x) sin x = x − x³ over 3! + x⁵ over 5! − …   (all x) cos x = 1 − x² over 2! + x⁴ over 4! − …   (all x) ln(1 + x) = x − x² over 2 + x³ over 3 − …   (−1 < x ≤ 1)

Near 0, a few terms give an excellent approximation. Far from 0, you need more terms, and for ln(1 + x) beyond x = 1 no number of terms will do.

Section XCommon mistakes

• Mixing up the sequence and the series. 1/n → 0, but Σ 1/n diverges.
• Using the nth-term test backwards. Terms going to 0 never proves convergence.
• Using a/(1 − r) when |r| ≥ 1. 1 + 2 + 4 + … does not equal −1.
• Starting at the wrong term. In a/(1 − r), a is the first term actually in the sum.
• Trusting a Taylor polynomial far from 0. The error grows as x moves away, and outside the interval of convergence more terms make it worse.

Section XIIIFrom Zeno to Maclaurin

Infinite sums puzzled people long before calculus. In the fifth century BC, Zeno of Elea posed paradoxes about motion, recorded by Aristotle in his Physics. In the Dichotomy, a runner must first reach the halfway point, then half of what is left, and so on without end; in the Achilles, the swift Achilles can never overtake a slower runner who has a head start. Hidden inside is the sum 1/2 + 1/4 + 1/8 + …, which we now know is exactly 1.

Around 1350 the French scholar Nicole Oresme proved that the harmonic series diverges, in his Questiones super geometriam Euclidis, by grouping the terms into blocks of 2, 4, 8, … terms, the same proof as in Section V. His Questiones circulated in manuscript and were not printed until the twentieth century.

In Kerala, in south India, Madhava of Sangamagrama found series equivalent to those for sin x, cos x and arctan x around 1400, more than two hundred years before they were found again in Europe. We know these results mainly through the works of later Kerala mathematicians, such as Jyesthadeva, whose Yukti-Bhasa, written in Malayalam around 1550, explains how the series were found.

In England, Brook Taylor described what we call Taylor’s theorem in a letter of 26 July 1712, and published it in his 1715 book Methodus incrementorum directa et inversa. He was not alone: James Gregory, Newton, Leibniz, Johann Bernoulli and de Moivre had all found versions of it. Its importance was not widely seen until 1772, when Lagrange called it the basic principle of the differential calculus.

The Scottish mathematician Colin Maclaurin used the special case centred at 0 in his two-volume Treatise of Fluxions (1742), and it is now called the Maclaurin series. Maclaurin did not claim it as his own discovery: he acknowledged Taylor’s work.

NotesSources and credits

• J. J. O’Connor and E. F. Robertson, MacTutor History of Mathematics, University of St Andrews: Zeno of Elea, on the Dichotomy and the Achilles as given in Aristotle’s Physics.
• MacTutor: Nicole Oresme (biography).
• J. Bell and V. Blåsjö, “Pietro Mengoli’s 1650 Proof that the Harmonic Series Diverges”, Mathematics Magazine 91(5), 2018, on Oresme’s proof in Question 2 of the Questiones super geometriam Euclidis, around 1350, first published in the 20th century.
• S. J. Kifowit and T. A. Stamps, “The Harmonic Series Diverges Again and Again”, The AMATYC Review, Spring 2006, on Oresme’s grouping proof.
• MacTutor: Madhava of Sangamagrama (biography), on his sine, cosine and arctangent series and Jyesthadeva’s Yukti-Bhasa.
• MacTutor: Brook Taylor (biography), on the 1712 letter to Machin, the 1715 Methodus incrementorum, earlier versions of the theorem and Lagrange in 1772.
• MacTutor: Colin Maclaurin (biography), on the 1742 Treatise of fluxions and his acknowledgement of Taylor.

Credits: every graph and diagram on this page is original SVG, computed from the actual functions, made for Jugami. The explorers run live in your browser with plain JavaScript, no outside libraries.