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Expansions like eˣ = 1 + x + x²/2 + ... are among the most useful tools in calculus and physics, and they are usually called Maclaurin series. Our post on Brook Taylor covers the general series and the small-angle shortcut sin θ ≈ θ. This post is about the Scottish mathematician whose name is on the special case at zero, why he wrote the book it appears in, and how to use the series without the usual mistakes.
Who was Colin Maclaurin?
The MacTutor biography records that he was born in February 1698 in Kilmodan, Argyllshire, where his father was the parish minister. His father died when Colin was six weeks old and his mother when he was nine, and he was brought up by an uncle. He entered the University of Glasgow at 11 and took his MA at 14, defending a thesis on the power of gravity.
In 1717, at 19, he was appointed professor of mathematics at Marischal College, Aberdeen, after ten days of competitive examinations. On a visit to London in 1719 he met Newton, writing that he was "very often" in his company, and was elected a Fellow of the Royal Society. In 1725, with Newton's written support, he joined the University of Edinburgh, where he taught for the rest of his life.
Why did he write the Treatise of Fluxions?
Bishop George Berkeley had attacked Newton's calculus as resting on unclear reasoning. Maclaurin's two-volume *Treatise of Fluxions* (1742), 763 pages long, was his reply. In the introduction, quoted by MacTutor, he wrote that Berkeley's objections "seemed to have been occasioned by the concise manner in which the elements of this method have been usually described", and that a fuller account was needed.
He tried to put calculus on the rigorous footing of Greek geometry and Archimedes' method of exhaustion. Judith Grabiner's study Was Newton's calculus a dead end? (1997) argues that the book influenced European mathematicians more than is usually thought, in topics including the fundamental theorem of calculus, maxima and minima, and the Euler-Maclaurin summation formula.
Did Maclaurin invent the Maclaurin series?
No, and he said so. MacTutor states that the series appears in the *Treatise of Fluxions* as a special case of Taylor's result, and that Maclaurin acknowledged Taylor's contribution. MacTutor also credits him with the integral test for the convergence of an infinite series, which he found about 50 years before Cauchy was born.
The series says that if you know a function and all its derivatives at 0, you can rebuild it nearby:
f(x) = f(0) + x f′(0) + (x²/2!) f″(0) + (x³/3!) f‴(0) + ...
Worked example 1: e⁰·¹ without a calculator
Every derivative of eˣ is eˣ, and e⁰ = 1, so eˣ = 1 + x + x²/2 + x³/6 + ...
x = 0.1: 1 + 0.1 = 1.1.
Add x²/2 = 0.01/2 = 0.005: 1.105.
Add x³/6 = 0.001/6 = 0.0001667: 1.1051667.
The true value is 1.1051709. Four terms are already correct to five decimal places.
Worked example 2: ln 1.1 and a limit
For ln(1 + x) the derivatives at 0 give ln(1 + x) = x − x²/2 + x³/3 − x⁴/4 + ...
x = 0.1: 0.1 − 0.005 + 0.000333 − 0.000025 = 0.095308.
The true value is 0.095310.
Series also settle limits. As x → 0, (eˣ − 1 − x)/x² becomes (x²/2 + x³/6 + ...)/x² = 1/2 + x/6 + ..., so the limit is 1/2.
What mistakes do students make with these series?
Using a series outside its range. The ln(1 + x) series only works for −1 < x ≤ 1. At x = 2 the terms 2 − 2 + 2.67 − 4 + ... grow without limit.
Dropping the factorials. The x³ term of eˣ is x³/6, not x³/3.
Stopping too soon in a limit. In (eˣ − 1 − x)/x², keeping only 1 + x gives 0/x², which looks like 0. You need the x² term.
Series expansions are not listed in the JEE (Main) 2026 syllabus, but they are a common way to check limit answers and approximations. If a limit keeps giving you a wrong answer, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
What can a student take from Maclaurin's way of working?
He took a public criticism seriously and answered it with a careful book rather than an insult. He credited Taylor openly. And he was, by every account MacTutor quotes, a patient teacher: if students could not follow a proof, he would suspect "his own explanation to have been obscure" and try again in another way. Alexander Carlyle called him "the clearest and most agreeable lecturer on that abstract science that ever I heard."
When you cannot follow your own solution a week later, treat it the way Maclaurin treated his lectures: rewrite it more clearly.
Try it now: use four terms of the series for cos x = 1 − x²/2 + x⁴/24 − ... to estimate cos 0.5, and compare with a calculator.
