Taylor Series Expansion

Taylor Series Expansion represents a sufficiently smooth function as an infinite sum of polynomial terms built from its derivatives at a single point. Key result in words: near a point

a

, a function behaves like a polynomial whose coefficients are

f^{(n)} (a) / n!

. Where it applies: approximation, numerical analysis, differential equations, physics, complex analysis. Who/when: named for Brook Taylor (1715), with roots in Gregory, Newton, and the Bernoullis; the centered-at-zero case bears Maclaurin's name.

The Taylor series of a function $f$ about a point $a$ is the power series $\sum_{n = 0}^{\infty} \frac{f ^{(n)} ( a )}{n !} (x - a)^{n}$ , which encodes the local behavior of $f$ entirely in terms of its derivatives at $a$ . It matters because polynomials are the simplest functions to evaluate, differentiate, and integrate, so replacing a complicated $f$ by its Taylor polynomial turns hard problems into algebra. The expansion is the engine behind calculator algorithms, perturbation theory, and the analysis of nearly every transcendental function. A surprising fact: a function can be infinitely differentiable yet have a Taylor series that converges everywhere to the wrong answer — smoothness alone does not guarantee a function equals its own Taylor series.

The Core Insight

A function that is "smooth enough" is locally indistinguishable from a polynomial. The first derivative captures slope, the second captures curvature, the third captures the rate of change of curvature, and so on. Each successive derivative refines the polynomial fit, and in the limit the polynomial of infinite degree may reproduce the function exactly within a region called the interval (or disk) of convergence.

Formal Statement

Taylor Series and Taylor Polynomial

Let $f$ be a real-valued function that is $n$ times differentiable at a point $a \in R$ . The $n$ -th order Taylor polynomial of $f$ about $a$ is

T_{n} (x) = k = 0 \sum n \frac{f ^{(k)} ( a )}{k !} (x - a)^{k} .

⟳demand that

T_{n}

and

f

share value and first

n

derivatives at

a

; matching

T_{n}^{(k)} (a) = f^{(k)} (a)

forces each coefficient to be

f^{(k)} (a) / k!

If $f$ is infinitely differentiable at $a$ , the Taylor series is the limiting power series

f (x) \sim n = 0 \sum \infty \frac{f ^{(n)} ( a )}{n !} (x - a)^{n} .

⟳take

n \to \infty

in [1]; the

\sim

symbol is used because equality requires the remainder to vanish.

When $a = 0$ this is called the Maclaurin series:

f (x) \sim n = 0 \sum \infty \frac{f ^{(n)} ( 0 )}{n !} x^{n} .

⟳substitute

a = 0

into [2].

Here $f^{(k)}$ denotes the $k$ -th derivative ( $f^{(0)} = f$ ), $k!$ is the factorial, $a$ is the center or base point, and $x - a$ is the displacement from the center.

Taylor's Theorem with Remainder

Let $f$ be $n + 1$ times differentiable on an open interval containing $a$ and $x$ . Then

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

⟳this is an exact equality, not an approximation;

R_{n}

absorbs every error.

where the Lagrange form of the remainder is

R_{n} (x) = \frac{f ^{(n + 1)} ( ξ )}{( n + 1 )!} (x - a)^{n + 1}

⟳apply the generalized Mean Value Theorem (Cauchy) repeatedly, or use Rolle's theorem on an auxiliary function;

ξ

lies strictly between

a

and

x

for some $ξ$ between $a$ and $x$ . The integral form is

R_{n} (x) = \frac{1}{n !} \int_{a}^{x} (x - t)^{n} f^{(n + 1)} (t) d t .

⟳integrate by parts

n

times starting from the Fundamental Theorem of Calculus

f (x) - f (a) = \int_{a}^{x} f^{'} (t) d t

The Taylor series of $f$ converges to $f (x)$ at a point $x$ if and only if $R_{n} (x) \to 0$ as $n \to \infty$ .

Why It Holds / Derivation

Taylor's Theorem via Repeated Integration by Parts

We establish the integral remainder [6], from which Lagrange's form [5] follows.

Step 1. By the Fundamental Theorem of Calculus,

f (x) = f (a) + \int_{a}^{x} f^{'} (t) d t .

⟳this is the

n = 0

base case; the integral is exactly

R_{0} (x)

Step 2. Integrate

\int_{a}^{x} f^{'} (t) d t

by parts choosing

u = f^{'} (t)

and

d v = d t

, but use the antiderivative

v = - (x - t)

(legal since

x

is constant in the integration):

\int_{a}^{x} f^{'} (t) d t = f^{'} (a) (x - a) + \int_{a}^{x} (x - t) f^{''} (t) d t .

⟳

[- (x - t) f^{'} (t)]_{a}^{x} = (x - a) f^{'} (a)

since the upper boundary term vanishes.

Step 3. Repeating with antiderivative

v = - \frac{( x - t ) ^{2}}{2}

on the new integral yields the next term

\frac{f ^{''} ( a )}{2} (x - a)^{2}

plus a remainder containing

f^{'''}

. Inductively, after

n

integrations by parts,

f (x) = k = 0 \sum n \frac{f ^{(k)} ( a )}{k !} (x - a)^{k} + \frac{1}{n !} \int_{a}^{x} (x - t)^{n} f^{(n + 1)} (t) d t .

⟳each step lowers the power on the remaining integral's

f^{(j)}

and raises the power of

(x - t)

, generating

1/ k!

factors.

Step 4. To recover the Lagrange form, apply the Mean Value Theorem for integrals: if

f^{(n + 1)}

is continuous, there exists

ξ

between

a

and

x

with

\frac{1}{n !} \int_{a}^{x} (x - t)^{n} f^{(n + 1)} (t) d t = \frac{f ^{(n + 1)} ( ξ )}{n !} \int_{a}^{x} (x - t)^{n} d t = \frac{f ^{(n + 1)} ( ξ )}{( n + 1 )!} (x - a)^{n + 1} .

⟳

(x - t)^{n}

has constant sign on the interval, so the weighted MVT pulls

f^{(n + 1)}

out at some interior point

ξ

■

Brook Taylor | 1685–1731

An English mathematician and member of the Royal Society, Brook Taylor published Methodus Incrementorum Directa et Inversa in 1715, containing the general theorem now bearing his name. Taylor did not address convergence and his statement was incomplete by modern standards; the importance of the series for the foundations of calculus was recognized only later by Lagrange, who in 1772 called it "the main foundation of differential calculus." Taylor also made early contributions to the calculus of finite differences, perspective in art, and the physics of vibrating strings.

Building Intuition

Think of building a function out of progressively finer corrections. Start at the point $a$ with the constant $f (a)$ — a flat line at the right height. Add $f^{'} (a) (x - a)$ to give the line the correct slope: this is the tangent line, the best linear fit. Add $\frac{1}{2} f^{''} (a) (x - a)^{2}$ to bend the line into the correct curvature: now you have the best parabolic fit. Each new term is a polynomial bump that fixes the next-finest mismatch in derivatives while leaving all lower derivatives at $a$ untouched (because $(x - a)^{k}$ and its first $k - 1$ derivatives vanish at $a$ ). The remainder $R_{n}$ measures how much "wiggle" of $f$ remains unmodeled, and it shrinks like $(x - a)^{n + 1}$ — so the approximation is excellent near $a$ and degrades as you move away.

Successive Taylor polynomials of sin(x) about a=0 converging to the curve

Worked Examples

Approximating e^{0.1} with a Maclaurin polynomial

Given:

f (x) = e^{x}

, centered at

a = 0

. We want

e^{0.1}

to within a guaranteed error bound. Step 1 — coefficients. Since

f^{(n)} (x) = e^{x}

, every

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

, so

e^{x} = n = 0 \sum \infty \frac{x ^{n}}{n !} = 1 + x + \frac{x ^{2}}{2} + \frac{x ^{3}}{6} + \dots .

⟳substitute

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

into the Maclaurin formula [3].

Step 2 — evaluate $T_{3}$ at $x = 0.1$ .

T_{3} (0.1) = 1 + 0.1 + \frac{0.01}{2} + \frac{0.001}{6} = 1.1051 \overline{6} .

⟳plug

x = 0.1

into [11] truncated at

n = 3

Step 3 — bound the error. By Lagrange [5] with

n = 3

ξ \in (0, 0.1)

∣ R_{3} ∣ = \frac{e ^{ξ}}{4 !} (0.1)^{4} \leq \frac{e ^{0.1}}{24} (1 0^{- 4}) < \frac{1.2}{24} \times 1 0^{- 4} = 5 \times 1 0^{- 6} .

⟳bound

e^{ξ}

e^{0.1} < 1.2

on the interval.

Answer:

e^{0.1} \approx 1.105167, error < 5 \times 1 0^{- 6}

(dimensionless). The true value is

1.1051709 \dots

, confirming the bound.

Curvature of a cosine and a small-angle limit

Given: A pendulum's restoring quantity involves

cos θ

. Approximate

cos θ

for small

θ

(radians) and evaluate at

θ = 0.2

rad. Step 1 — Maclaurin series of cosine. Derivatives cycle

cos, - sin, - cos, sin

, giving values

1, 0, - 1, 0

0

cos θ = 1 - \frac{θ ^{2}}{2 !} + \frac{θ ^{4}}{4 !} - \dots .

⟳only even derivatives survive; alternating signs from the second-derivative sign flips.

Step 2 — second-order approximation. Keep through

θ^{2}

cos (0.2) \approx 1 - \frac{( 0.2 ) ^{2}}{2} = 1 - 0.02 = 0.98.

⟳truncate [14] after the quadratic term.

Step 3 — error from the next term. The leading omitted term is

θ^{4} /24 = (0.2)^{4} /24 = 6.67 \times 1 0^{- 5}

. Answer:

cos (0.2) \approx 0.98, error \approx 6.7 \times 1 0^{- 5}

(dimensionless). The small-angle approximation

cos θ \approx 1 - θ^{2} /2

is now justified quantitatively.

Common Misconceptions

Infinitely differentiable does NOT mean equal to its Taylor series. Existence of all derivatives at

a

guarantees the Taylor series can be written down, but convergence of the series to

f

requires

R_{n} (x) \to 0

. Functions for which the series converges to

f

on a neighborhood are called analytic; not every smooth (

C^{\infty}

) function is analytic. Always check the remainder, not just differentiability. The interval of convergence is not always the whole real line. The geometric series

\frac{1}{1 - x} = \sum x^{n}

converges only for

∣ x ∣ < 1

, even though

\frac{1}{1 - x}

is perfectly well-defined and smooth at, say,

x = 2

. The series cannot "see" past the singularity at

x = 1

; the radius of convergence is set by the distance to the nearest singularity in the complex plane.

Cauchy's flat function

Define

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

⟳the exponential decays faster than any power of

x

approaches zero.

Every derivative at the origin equals zero: $f^{(n)} (0) = 0$ for all $n$ . Therefore its Maclaurin series is identically $\sum 0 \cdot x^{n} = 0$ , which converges everywhere — but to the zero function, not to $f$ , since $f (x) > 0$ for $x \neq = 0$ . This $f$ is $C^{\infty}$ everywhere yet analytic nowhere on any neighborhood of $0$ . It is the canonical proof that smoothness $\neq =$ analyticity.

Connections

The Taylor series sits at the crossroads of analysis. It generalizes the Mean Value Theorem (the $n = 0$ case of the remainder) and the Linear Approximation (the $n = 1$ tangent-line case). Its convergence theory is the subject of Power Series and the Radius of Convergence, computed via the ratio or root tests from Convergence Tests for Series. In the complex domain it merges with the Laurent Series and is implied by Cauchy's Integral Formula, where analyticity of a complex differentiable function is automatic. The remainder term connects directly to Numerical Integration error estimates and to the Asymptotic Expansion, its divergent cousin used in physics.

Real-World Applications

Calculator and CPU function evaluation. Hardware and libraries compute $sin$ , $cos$ , $exp$ , and $ln$ using truncated Taylor/minimax polynomials with range reduction.
Physics small-angle pendulum. $sin θ \approx θ$ linearizes the equation of motion into simple harmonic motion, the foundation of clock design and the seconds pendulum.
Special relativity. Expanding $γ m c^{2} = m c^{2} (1 - v^{2} / c^{2})^{- 1/2}$ recovers the classical kinetic energy $\frac{1}{2} m v^{2}$ as the leading correction beyond rest energy.
Numerical methods. Euler's method, Runge–Kutta schemes, and finite-difference stencils derive their order of accuracy directly from Taylor expansions of the solution.
Financial engineering. The Greeks "delta" and "gamma" in option pricing are first- and second-order Taylor coefficients of the option value with respect to the underlying price.
Error propagation in metrology. Uncertainty in a measured quantity $f (x)$ is estimated via $δ f \approx ∣ f^{'} (x) ∣ δ x$ , the first-order Taylor term.
Perturbation theory in quantum mechanics. Energy levels are expanded in a small coupling parameter $λ$ , a Taylor series in the perturbation strength.

Historical Development

1668–1671 — James Gregory discovers series for $arctan$ , $tan$ , $sec$ and effectively the general Taylor expansion in his manuscripts, decades before Taylor.
1670s — Isaac Newton develops the binomial series and infinite-series methods extensively but publishes little.
1694 — Johann Bernoulli derives an integral-form series equivalent to Taylor's theorem.
1715 — Brook Taylor publishes the general theorem in Methodus Incrementorum, giving it his name.
1742 — Colin Maclaurin popularizes the $a = 0$ special case in his Treatise of Fluxions, attributing it to Taylor.
1772 — Joseph-Louis Lagrange champions the series as the foundation of calculus and introduces the remainder form named for him.
1797 — Lagrange's Théorie des fonctions analytiques attempts (incorrectly) to base all of calculus on Taylor series, exposing the need for rigor.
1823 — Augustin-Louis Cauchy provides rigorous convergence theory and constructs the flat function $e^{- 1/ x^{2}}$ showing smoothness $\neq =$ analyticity.
1840s — Cauchy and Weierstrass develop complex power series, explaining real radii of convergence via complex singularities.

Subtleties & Limits

Several deep issues lie just beyond the undergraduate boundary. Divergent but useful series: many physically important Taylor/asymptotic series (e.g., the perturbation series of quantum electrodynamics, or Stirling's series) diverge for every nonzero argument, yet their partial sums give superb finite-order approximations; the rigorous theory of Borel summation and resurgence rescues meaning from such divergence and remains an active research area. Quasianalytic classes: the Denjoy–Carleman theorem precisely characterizes which families of $C^{\infty}$ functions are determined by their derivatives at a point, sharpening the analytic/smooth divide. Optimal truncation: for asymptotic series there is a least-error truncation point, and understanding the "superasymptotic" and "hyperasymptotic" corrections beyond it is ongoing. Finally, multivariate and operator-valued Taylor expansions raise convergence and domain questions still studied in functional analysis.