Maclaurin Series

The Maclaurin series is the Taylor series of a function expanded about the point $x = 0$ : it represents a function $f (x)$ as an infinite power series $\sum_{n = 0}^{\infty} \frac{f ^{(n)} ( 0 )}{n !} x^{n}$ . Useful for approximating functions near the origin, evaluating limits, and deriving identities. Named after Colin Maclaurin (18th century), though the technique predates him.

The Maclaurin series of a function $f$ is the special case of the Taylor series centered at the origin. If $f$ is infinitely differentiable at $0$ , its Maclaurin series is the power series whose coefficients are determined by the derivatives of $f$ evaluated at $0$ . It matters because it converts transcendental functions into polynomials that can be added, differentiated, integrated, and computed term-by-term — making it the backbone of numerical analysis, physics approximations, and complex analysis. A surprising fact: a function can be infinitely differentiable at $0$ , possess a perfectly well-defined Maclaurin series, and yet that series can fail to equal the function anywhere except at $0$ itself (e.g. $e^{- 1/ x^{2}}$ ).

The Core Insight

Near a point, a smooth function "looks like" its tangent line; looking closer, it looks like a parabola; closer still, a cubic — and so on. The Maclaurin series captures all these progressively better polynomial approximations at once by matching every derivative of $f$ at $x = 0$ . The $n$ -th coefficient $a_{n} = f^{(n)} (0) / n!$ is precisely chosen so that the series and $f$ agree in value and in all derivatives at the origin.

Formal Statement

Maclaurin Series

Let $f$ be a function that is infinitely differentiable in a neighborhood of $x = 0$ . The Maclaurin series of $f$ is the power series

f (x) \sim n = 0 \sum \infty \frac{f ^{(n)} ( 0 )}{n !} x^{n} = f (0) + f^{'} (0) x + \frac{f ^{''} ( 0 )}{2 !} x^{2} + \frac{f ^{'''} ( 0 )}{3 !} x^{3} + \dots

⟳Specialize the Taylor series

\sum \frac{f ^{(n)} ( a )}{n !} (x - a)^{n}

to the center

a = 0

Here:

$f^{(n)} (0)$ is the $n$ -th derivative of $f$ evaluated at $0$ (with $f^{(0)} = f$ ),
$n!$ is the factorial of $n$ (with $0! = 1$ ),
$x^{n}$ is the $n$ -th power of the variable,
the symbol $\sim$ signals that we have associated a series to $f$ — equality holds only where the series converges to $f$ .

Convergence and Taylor's Theorem with Remainder

Let $f$ be $(N + 1)$ -times differentiable on an interval containing $0$ and $x$ . Then

f (x) = n = 0 \sum N \frac{f ^{(n)} ( 0 )}{n !} x^{n} + R_{N} (x),

⟳This is Taylor's theorem; the partial sum is the degree-

N

Maclaurin polynomial.

where the Lagrange form of the remainder is

R_{N} (x) = \frac{f ^{(N + 1)} ( ξ )}{( N + 1 )!} x^{N + 1}, ξ strictly between 0 and x .

⟳Apply the Cauchy mean value theorem / generalized Rolle's theorem to the remainder function;

ξ

exists by the intermediate-value behavior of derivatives.

The Maclaurin series converges to $f (x)$ at a point $x$ if and only if $N \to \infty lim R_{N} (x) = 0$ .

Why It Holds / Derivation

Why the coefficients must be $f^{(n)}(0)/n!$

Suppose $f$ can be represented near $0$ by some convergent power series:

f (x) = n = 0 \sum \infty a_{n} x^{n} = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \dots

⟳Assume term-by-term differentiation is valid inside the radius of convergence — a theorem for power series.

Step 1. Evaluate at

x = 0

. Every term with

x^{n}

(

n \geq 1

) vanishes, leaving

f (0) = a_{0}

. Step 2. Differentiate [4] term by term:

f^{'} (x) = a_{1} + 2 a_{2} x + 3 a_{3} x^{2} + \dots = n = 1 \sum \infty n a_{n} x^{n - 1} .

⟳Power series may be differentiated term-by-term within their open interval of convergence.

Setting $x = 0$ gives $f^{'} (0) = a_{1}$ .

Step 3. Differentiate again:

f^{''} (x) = 2 a_{2} + 6 a_{3} x + 12 a_{4} x^{2} + \dots = n = 2 \sum \infty n (n - 1) a_{n} x^{n - 2} .

⟳Repeat term-by-term differentiation; the constant and linear terms drop.

At $x = 0$ , $f^{''} (0) = 2 a_{2}$ , so $a_{2} = f^{''} (0) /2!$ .

Step 4. By induction, differentiating

n

times and setting

x = 0

kills all but the constant term, which equals

n! a_{n}

f^{(n)} (0) = n! a_{n} ⟹ a_{n} = \frac{f ^{(n)} ( 0 )}{n !} .

⟳The

n

-th derivative of

a_{k} x^{k}

0

0

unless

k = n

, in which case it is

n! a_{n}

Thus the coefficients are forced — if a power-series representation exists, it must be the Maclaurin series. $■$

Colin Maclaurin | 1698–1746

A Scottish mathematician and a child prodigy who entered the University of Glasgow at age eleven and held a professorship at Aberdeen by nineteen. He published the series bearing his name in his 1742 Treatise of Fluxions, a rigorous defense of Newton's calculus against George Berkeley's philosophical attacks. Maclaurin freely credited the more general result to Brook Taylor, presenting the $a = 0$ case as a useful special tool. He also made deep contributions to the theory of the gravitational attraction of ellipsoids, geometry, and actuarial science.

Building Intuition

Think of the Maclaurin polynomial of degree $N$ as the "best possible" polynomial of that degree for matching $f$ at the single point $x = 0$ . The constant term pins the height, the linear term pins the slope, the quadratic term pins the curvature, and each successive term corrects one more order of local shape. Adding more terms is like progressively sharpening a lens focused on the origin — the approximation hugs the true curve over an ever-wider window. The factorial in the denominator reflects how much "information" each derivative carries: high derivatives grow fast, but dividing by $n!$ tames them so the series can converge.

Successive Maclaurin polynomials of $\sin x$ converging to the curve near the origin

Worked Examples

Maclaurin series of $e^x$ and approximating $e^{0.2}$

Given:

f (x) = e^{x}

. We want its Maclaurin series and a numerical estimate of

e^{0.2}

. Step 1 — Derivatives at 0. Since

\frac{d}{d x} e^{x} = e^{x}

, every derivative is

e^{x}

, so

f^{(n)} (0) = e^{0} = 1

for all

n

. Step 2 — Build the series. Coefficients are

a_{n} = 1/ n!

e^{x} = n = 0 \sum \infty \frac{x ^{n}}{n !} = 1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots, valid for all x \in R .

⟳Use

f^{(n)} (0) = 1

in equation [1]; the ratio test gives infinite radius of convergence.

Step 3 — Evaluate at $x = 0.2$ through 4 terms:

1 + 0.2 + \frac{0.04}{2} + \frac{0.008}{6} = 1 + 0.2 + 0.02 + 0.001333 = 1.221333 \dots

Step 4 — Error bound. With

N = 3

R_{3} = \frac{e ^{ξ}}{4 !} (0.2)^{4} \leq \frac{e ^{0.2}}{24} (0.0016) \approx 8.1 \times 1 0^{- 5}

e^{0.2} \approx 1.22133 (true value 1.221403, error < 1 0^{- 4})

Maclaurin series of $\cos x$ and a limit

Given:

f (x) = cos x

. Find the series and use it to evaluate

x \to 0 lim \frac{1 - cos x}{x ^{2}}

. Step 1 — Derivatives at 0. Cyclic:

cos, - sin, - cos, sin, \dots

0

give

1, 0, - 1, 0, 1, \dots

Odd derivatives vanish. Step 2 — Series:

cos x = n = 0 \sum \infty \frac{( - 1 ) ^{n}}{( 2 n )!} x^{2 n} = 1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \frac{x ^{6}}{6 !} + \dots

⟳Only even powers survive; signs alternate from the derivative cycle, giving the

(- 1)^{n} / (2 n)!

pattern.

Step 3 — Substitute into the limit.

1 - cos x = \frac{x ^{2}}{2 !} - \frac{x ^{4}}{4 !} + \dots \Rightarrow \frac{1 - cos x}{x ^{2}} = \frac{1}{2} - \frac{x ^{2}}{24} + \dots

Step 4 — Take $x \to 0$ : all terms with

x

vanish.

x \to 0 lim \frac{1 - cos x}{x ^{2}} = \frac{1}{2}

Common Misconceptions

A function is not always equal to its Maclaurin series. Three independent things can go wrong:

The series may have a finite radius of convergence $R$ , so it equals $f$ only on $(- R, R)$ . Example: $\frac{1}{1 - x} = \sum x^{n}$ converges only for $∣ x ∣ < 1$ , even though $f$ is smooth at $x = 2$ .
The series may converge but to the wrong value (see the counterexample below).
Existence of all derivatives at $0$ is required — $∣ x ∣$ and $∣ x ∣$ are not differentiable at $0$ , so they have no Maclaurin series.

Also do not confuse the Maclaurin polynomial (a finite truncation, always a polynomial) with the Maclaurin series (the infinite object). Convergence questions only arise for the series.

A smooth function whose Maclaurin series is identically zero

Define

f (x) = {e^{- 1/ x^{2}}, 0, x \neq = 0, x = 0.

⟳Repeated differentiation produces

P (1/ x) e^{- 1/ x^{2}}

for polynomials

P

; the exponential decay dominates, forcing each derivative to

0

at the origin.

This function is infinitely differentiable everywhere, and every derivative at $0$ equals $0$ : $f^{(n)} (0) = 0$ for all $n$ . Hence its Maclaurin series is $0 + 0 \cdot x + 0 \cdot x^{2} + \dots = 0$ , which converges everywhere — but to the zero function, not to $f$ , since $f (x) > 0$ for $x \neq = 0$ . A function equal to its Maclaurin series on a neighborhood is called real-analytic; this $f$ is smooth but not analytic at $0$ .

Connections

The Maclaurin series is the $a = 0$ instance of the Taylor Series, inheriting all its convergence machinery. The coefficient extraction relies on Term-by-Term Differentiation of Power Series and the underlying theory of the Radius of Convergence (via the ratio and root tests). In Complex Analysis, a function holomorphic on a disk centered at $0$ automatically equals its Maclaurin series there — the analytic pathology of the real counterexample cannot occur, revealing a deep rigidity. The series also connects to Generating Functions in combinatorics, where coefficients encode counting sequences, and to L'Hôpital's Rule as an alternative method for evaluating indeterminate limits.

Real-World Applications

Computer arithmetic libraries: CPU and software routines compute $sin$ , $cos$ , $exp$ , and $ln$ by argument reduction followed by short truncated Maclaurin (or related minimax) polynomials.
Physics — small-oscillation approximation: The pendulum equation uses $sin θ \approx θ$ (first Maclaurin term) to linearize into simple harmonic motion.
Special relativity: Kinetic energy $E = m c^{2} (γ - 1)$ expands as $\frac{1}{2} m v^{2} + \frac{3}{8} m \frac{v ^{4}}{c ^{2}} + \dots$ , recovering Newtonian energy as the leading term.
Engineering — signal processing: Maclaurin/Taylor expansions linearize nonlinear transfer functions around an operating point for stability analysis.
Numerical ODE solvers: Taylor-series methods and the derivation of Runge–Kutta schemes match Maclaurin coefficients of the exact solution.
Finance: The price sensitivity of bonds (duration and convexity) is a second-order Maclaurin expansion of price in yield.
Optics: The paraxial (Gaussian) approximation uses $sin θ \approx θ$ and $cos θ \approx 1 - θ^{2} /2$ to derive lens equations.

Historical Development

c. 1665–1671: Isaac Newton derives power-series expansions for $sin$ , $cos$ , $arcsin$ , and the binomial series, computing with them freely though without modern convergence theory.
1668: Nicolaus Mercator publishes the series for $ln (1 + x)$ — an early named special expansion.
1671: James Gregory independently obtains Taylor-type expansions, including the series for $arctan x$ .
1715: Brook Taylor publishes the general Taylor theorem in Methodus Incrementorum Directa et Inversa.
1730s: Continental mathematicians, notably Euler, exploit such expansions for myriad results (e.g. the Basel problem solution and Euler's formula).
1742: Colin Maclaurin presents the $a = 0$ special case prominently in his Treatise of Fluxions, after which it bears his name.
1755–1770s: Euler systematizes power-series manipulation in his Introductio and Institutiones, treating series almost as polynomials.
1797–1813: Lagrange formalizes the remainder term, attempting to base all of calculus on Taylor series (Théorie des fonctions analytiques).
1821–1823: Cauchy provides rigorous convergence theory, defines analyticity, and gives the famous $e^{- 1/ x^{2}}$ counterexample showing smoothness does not imply analyticity.

Subtleties & Limits

The boundary between smooth ( $C^{\infty}$ ) and real-analytic functions remains conceptually rich. While the $e^{- 1/ x^{2}}$ example shows the gap, classifying exactly which smooth functions are analytic, and characterizing the growth of derivatives (the Denjoy–Carleman theorem delineates "quasi-analytic" classes between the two), occupies a deep corner of analysis. In several variables and on manifolds, questions about analytic continuation, Borel's theorem (every sequence of values is realizable as the derivatives of some smooth function), and the resurgence/Borel-summation of divergent asymptotic Maclaurin series (common in physics, e.g. perturbative QED) are active research areas. Determining the exact radius of convergence or asymptotic behavior of series coefficients for implicitly or recursively defined functions can be genuinely hard.

Maclaurin Series

The Core Insight

Formal Statement

Maclaurin Series

Convergence and Taylor's Theorem with Remainder

Why It Holds / Derivation

Why the coefficients must be $f^{(n)}(0)/n!$

Colin Maclaurin | 1698–1746

Building Intuition

Successive Maclaurin polynomials of $\sin x$ converging to the curve near the origin

Worked Examples

Maclaurin series of $e^x$ and approximating $e^{0.2}$

Maclaurin series of $\cos x$ and a limit

Common Misconceptions

A smooth function whose Maclaurin series is identically zero

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also

Before this

Where this leads