Taylor's Theorem and Error Bounds

Taylor's Theorem states that a sufficiently differentiable function can be approximated near a point by a polynomial built from its derivatives, with an explicit, controllable error term (the remainder). The key result:

f (x) = P_{n} (x) + R_{n} (x)

, where

P_{n}

is the degree-

n

Taylor polynomial and

R_{n}

is bounded by the size of the

(n + 1)

-th derivative. It applies wherever local polynomial approximation, error estimation, or series expansion is needed — numerical analysis, physics, engineering. Named for Brook Taylor (1715), with the remainder forms due to Lagrange, Cauchy, and Schlömilch (18th–19th c.).

Taylor's Theorem is the rigorous foundation for approximating functions by polynomials. Given a function $f$ that is $n + 1$ times differentiable on an interval containing a base point $a$ , the theorem guarantees that $f (x)$ equals its $n$ -th degree Taylor polynomial $P_{n} (x)$ plus a remainder $R_{n} (x)$ whose exact form is given explicitly (Lagrange, Cauchy, or integral). This matters because it converts the vague idea "smooth functions look like polynomials nearby" into a precise inequality: it tells you how good the approximation is and how fast the error shrinks as $x \to a$ . The surprising fact is that even a function having a complete Taylor series at every point need not equal that series anywhere except at the base point — smoothness alone does not guarantee a function is recoverable from its derivatives.

The Core Insight

The value, slope, curvature, and all higher derivatives of $f$ at a single point $a$ encode enough information to reconstruct $f$ approximately throughout a neighborhood. Matching the first $n + 1$ derivatives at $a$ forces the approximating polynomial to "hug" the curve, and the leftover error is governed entirely by the next derivative — the one we did not match. The remainder is not a vague fudge factor; it is a genuine equality of the form $R_{n} (x) = \frac{f ^{(n + 1)} ( ξ )}{( n + 1 )!} (x - a)^{n + 1}$ for some unknown $ξ$ between $a$ and $x$ .

Formal Statement

Taylor Polynomial

Let $f$ be $n$ times differentiable at $a$ . The Taylor polynomial of degree $n$ of $f$ centered at $a$ is

P_{n} (x) = k = 0 \sum n \frac{f ^{(k)} ( a )}{k !} (x - a)^{k} = f (a) + f^{'} (a) (x - a) + \dots + \frac{f ^{(n)} ( a )}{n !} (x - a)^{n} .

⟳it is the unique degree-

\leq n

polynomial

P

satisfying

P^{(k)} (a) = f^{(k)} (a)

for

k = 0, \dots, n

; differentiate the sum

k

times and evaluate at

x = a

— only the

k

-th term survives.

Here $f^{(k)} (a)$ is the $k$ -th derivative of $f$ at $a$ , $k!$ is the factorial, and $(x - a)^{k}$ measures displacement from the base point $a$ . When $a = 0$ this is called the Maclaurin polynomial. The remainder is defined by $R_{n} (x) = f (x) - P_{n} (x)$ .

Taylor's Theorem with Lagrange Remainder

Let $f : [a, x] \to R$ (or $[x, a]$ ) be such that $f, f^{'}, \dots, f^{(n)}$ are continuous on the closed interval and $f^{(n + 1)}$ exists on the open interval. Then there exists a point $ξ$ strictly between $a$ and $x$ such that

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

⟳apply the Cauchy Mean Value Theorem (or repeated Rolle's Theorem) to an auxiliary function chosen so its zeros pin down

ξ

; see the proof below.

The final term is the Lagrange form of the remainder, $R_{n} (x) = \frac{f ^{(n + 1)} ( ξ )}{( n + 1 )!} (x - a)^{n + 1}$ .

Other Forms of the Remainder

For $f^{(n + 1)}$ continuous, the integral (exact) form is

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

⟳start from

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

and integrate by parts repeatedly, each time differentiating the

(x - t)

power and integrating the derivative factor.

The Cauchy form (for some $θ \in (0, 1)$ , with $ξ = a + θ (x - a)$ ) is

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

⟳apply the ordinary Mean Value Theorem (not Cauchy's generalized one) to the integral form, or use the auxiliary function with a linear weight instead of a power weight.

Peano Remainder — Qualitative Form

If only $f^{(n)} (a)$ exists (no $(n + 1)$ -th derivative required), then

f (x) = P_{n} (x) + o ((x - a)^{n}) as x \to a,

⟳subtract

P_{n}

, apply L'Hôpital's rule

n

times to

R_{n} (x) / (x - a)^{n}

, using the matching of all derivatives up to order

n

meaning $lim_{x \to a} R_{n} (x) / (x - a)^{n} = 0$ . This is the weakest, purely local statement and underlies asymptotic analysis.

Why It Holds / Derivation

Lagrange Remainder via Auxiliary Function

Goal: Find

ξ

between

a

and

x

with

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

. Fix

x

and treat

a

(renamed

t

) as the variable.

Define an auxiliary function. For $t$ in the closed interval, set

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

Then

g (x) = 0

(only the

k = 0

term survives, giving

f (x) - f (x)

) and

g (a) = R_{n} (x)

Differentiate $g$ . Each term's derivative telescopes by the product rule:

g^{'} (t) = - \frac{f ^{(n + 1)} ( t )}{n !} (x - t)^{n} .

All intermediate terms cancel pairwise — this is the engine of the proof.

Define a comparison function. Let $h (t) = (\frac{x - t}{x - a})^{n + 1}$ , so $h (a) = 1$ , $h (x) = 0$ , and $h^{'} (t) = - \frac{( n + 1 ) ( x - t ) ^{n}}{( x - a ) ^{n + 1}}$ .

Apply Cauchy's Mean Value Theorem to $g$ and $h$ on $[a, x]$ : there is $ξ$ between $a$ and $x$ with

\frac{g ( x ) - g ( a )}{h ( x ) - h ( a )} = \frac{g ^{'} ( ξ )}{h ^{'} ( ξ )} .

Substitute the endpoint values. The left side is $\frac{0 - R _{n} ( x )}{0 - 1} = R_{n} (x)$ . The right side simplifies:

\frac{g ^{'} ( ξ )}{h ^{'} ( ξ )} = \frac{- \frac{f ^{(n + 1)} ( ξ )}{n !} ( x - ξ ) ^{n}}{- \frac{( n + 1 ) ( x - ξ ) ^{n}}{( x - a ) ^{n + 1}}} = \frac{f ^{(n + 1)} ( ξ )}{( n + 1 )!} (x - a)^{n + 1} .

Conclude. Equating both sides gives the Lagrange remainder. $■$

Brook Taylor | 1685–1731

An English mathematician and member of the Royal Society, Taylor published the formula bearing his name in his 1715 treatise Methodus Incrementorum Directa et Inversa. He derived it as a limit of Newton's interpolation formula and used it to study vibrating strings and the motion of projectiles — though without any notion of convergence or remainder. The crucial error term came a generation later from Joseph-Louis Lagrange (1797), and the rigorous treatment with explicit remainders is largely the work of Cauchy in the 1820s. Taylor also founded the mathematical theory of linear perspective and contributed to the calculus of finite differences.

Building Intuition

Think of building an approximation one derivative at a time. Start with a flat horizontal line at height $f (a)$ — it gets the value right but ignores slope. Add a tilt $f^{'} (a) (x - a)$ — now the tangent line matches direction. Add curvature $\frac{f ^{''} ( a )}{2} (x - a)^{2}$ — now you bend the right way. Each new term corrects the highest unmatched derivative, and the error you have left is dominated by the first derivative you stopped matching, scaled by $(x - a)^{n + 1} / (n + 1)!$ . Two forces fight in the remainder: the factorial in the denominator crushes the error (it grows astronomically), while $(x - a)^{n + 1}$ explodes if $∣ x - a ∣ > 1$ . Near the base point, the factorial wins decisively, which is why Taylor polynomials are local tools — superb close to $a$ , often useless far away.

Degree-1, -2, and -3 Taylor approximations of a curve at base point $a$, with the error (remainder) shown as the vertical gap.

Worked Examples

Estimating $e^{0.1}$ with a Maclaurin Polynomial

Given: Approximate

e^{0.1}

using the degree-2 Maclaurin polynomial of

f (x) = e^{x}

, and bound the error. Setup. Here

a = 0

x = 0.1

. Since

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

, all derivatives at

0

equal

1

P_{2} (x) = 1 + x + \frac{x ^{2}}{2} .

Compute the approximation.

P_{2} (0.1) = 1 + 0.1 + \frac{0.01}{2} = 1.105.

Bound the error (Lagrange). With

n = 2

R_{2} (0.1) = \frac{f ^{(3)} ( ξ )}{3 !} (0.1)^{3} = \frac{e ^{ξ}}{6} (0.001), 0 < ξ < 0.1.

Since

e^{ξ} < e^{0.1} < 1.2

∣ R_{2} (0.1) ∣ < \frac{1.2}{6} (0.001) = 0.0002.

Answer.

e^{0.1} \approx 1.105, error < 2 \times 1 0^{- 4}

(The true value

e^{0.1} = 1.10517 \dots

, so the actual error is

\approx 1.7 \times 1 0^{- 4}

— consistent with the bound.) Units are dimensionless.

How Many Terms for $\sin(0.5)$ to $10^{-6}$?

Given: Find the degree

n

of the Maclaurin polynomial of

sin x

needed to compute

sin (0.5)

with error below

1 0^{- 6}

. Setup. For

sin x

, every derivative satisfies

∣ f^{(n + 1)} (ξ) ∣ \leq 1

. The Lagrange bound is

∣ R_{n} (0.5) ∣ \leq \frac{∣0.5 ∣ ^{n + 1}}{( n + 1 )!} .

Test values.

| $n + 1$ | $(0.5)^{n + 1} / (n + 1)!$ | |---|---| | 4 | $0.0625/24 \approx 2.6 \times 1 0^{- 3}$ | | 6 | $0.015625/720 \approx 2.2 \times 1 0^{- 5}$ | | 8 | $0.00390625/40320 \approx 9.7 \times 1 0^{- 8}$ |

So $n + 1 = 8$ , i.e. $n = 7$ , gives error $< 1 0^{- 6}$ . (Because $sin$ has only odd-power terms, this means using terms up to $x^{7}$ .)

Compute.

P_{7} (0.5) = 0.5 - \frac{0. 5 ^{3}}{6} + \frac{0. 5 ^{5}}{120} - \frac{0. 5 ^{7}}{5040} \approx 0.479425539.

Answer.

n = 7 suffices; sin (0.5) \approx 0.4794255, error < 1 0^{- 6}

(True value

sin 0.5 = 0.4794255386 \dots

Common Misconceptions

"Infinitely differentiable" does NOT imply "equals its Taylor series." Many students assume that if all derivatives exist, the Taylor series converges to the function. Convergence of the series and the series summing to

f

are separate questions — the remainder

R_{n} (x)

must tend to

0

n \to \infty

, and this can fail even when the series converges to a different value. Also beware: the point

ξ

in the Lagrange remainder is unknown — you do not get to choose it. You can only bound

∣ f^{(n + 1)} ∣

over the whole interval to get a usable error estimate.

A Smooth Function Whose Taylor Series Is Useless

Consider

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

This function is

C^{\infty}

(infinitely differentiable) everywhere, and one can show by induction that

f^{(k)} (0) = 0

for every

k

. Therefore its Maclaurin series is identically

0

k = 0 \sum \infty \frac{f ^{(k)} ( 0 )}{k !} x^{k} = 0 + 0 \cdot x + 0 \cdot x^{2} + \dots = 0.

The series converges everywhere — but to the zero function, not to

f

, which is positive for all

x \neq = 0

. The remainder

R_{n} (x) = f (x)

never shrinks. This shows smoothness is not enough; such non-analytic functions are central to the study of Analytic Functions.

Connections

Taylor's Theorem sits at the junction of several core ideas. It generalizes the Mean Value Theorem — indeed the case $n = 0$ of equation [2] is the Mean Value Theorem $f (x) = f (a) + f^{'} (ξ) (x - a)$ . Letting $n \to \infty$ leads to Taylor Series and the theory of Power Series, where the radius of convergence governs validity. The integral remainder [3] is a direct consequence of Integration by Parts and the Fundamental Theorem of Calculus. In several variables it generalizes to expansions involving the Hessian Matrix and higher tensors, foundational to Optimization. The Peano form [5] is the bridge to Asymptotic Expansions and the theory of Big-O Notation used throughout numerical analysis.

Real-World Applications

Numerical integration (Simpson's & trapezoidal rules): Error terms like $- \frac{( b - a ) ^{5}}{2880} f^{(4)} (ξ)$ for Simpson's rule come directly from Taylor expansion of the integrand.
Newton's method for root-finding: Quadratic convergence is proved by expanding $f$ to second order around the root and bounding the remainder.
Physics — small-oscillation / harmonic approximation: Expanding a potential $V (x)$ to second order about equilibrium yields $V \approx V_{0} + \frac{1}{2} V^{''} (x_{0}) (x - x_{0})^{2}$ , the simple harmonic oscillator.
Special relativity: The kinetic energy $K = m c^{2} (γ - 1)$ expanded for small $v / c$ recovers $\frac{1}{2} m v^{2}$ plus relativistic corrections of order $v^{4} / c^{2}$ .
Computer math libraries (CORDIC, glibc): Functions like $sin$ , $exp$ , $lo g$ are evaluated via Taylor/Chebyshev polynomials with rigorously bounded remainders to guarantee last-bit accuracy.
Finite-difference methods in PDE solvers: Truncation error of schemes (e.g. $O (Δ x^{2})$ central differences) is read off from Taylor remainders.
Engineering linearization: Control systems and circuit analysis linearize nonlinear devices around an operating point using the first-order Taylor term, with the remainder quantifying distortion.

Historical Development

1671 — James Gregory discovers special cases of Taylor series (for $arctan$ , $tan$ , $sec$ ) decades before Taylor, unpublished in full.
1715 — Brook Taylor publishes the general formula in Methodus Incrementorum, derived via finite differences, with no remainder or convergence theory.
1742 — Colin Maclaurin popularizes the special case $a = 0$ in his Treatise of Fluxions; the name "Maclaurin series" sticks despite his crediting Taylor.
1772 — Lagrange begins promoting Taylor expansions as the foundation of calculus, hoping to eliminate limits and infinitesimals.
1797 — Lagrange introduces the explicit remainder term in Théorie des fonctions analytiques, giving the Lagrange form.
1821 — Augustin-Louis Cauchy, in Cours d'analyse and later works, supplies rigorous definitions of limits, derives the Cauchy form of the remainder, and exposes that the series need not equal the function.
1823 — Cauchy publishes the $e^{- 1/ x^{2}}$ counterexample, demolishing Lagrange's program of founding calculus on Taylor series alone.
1880s — Oskar Schlömilch & Roché give the general (parametrized) remainder form unifying the Lagrange and Cauchy versions.
20th c. — Giuseppe Peano formalizes the qualitative $o ((x - a)^{n})$ remainder, central to modern asymptotic analysis.

Subtleties & Limits

The boundary between "has a Taylor series" and "equals its Taylor series" defines the class of real-analytic functions, and characterizing this class intrinsically is subtle. A deep result (Pringsheim, Borel) shows that for any prescribed sequence of real numbers ${a_{k}}$ — no matter how wildly it grows — there exists a $C^{\infty}$ function with $f^{(k)} (0) = a_{k}$ (Borel's theorem). Thus the Taylor coefficients place essentially no constraint on a smooth function. Quantifying how a smooth-but-non-analytic function's remainder behaves leads to the theory of Gevrey classes and quasi-analytic functions (Denjoy–Carleman theorem), which gives sharp conditions on the growth of $∣ f^{(n)} ∣$ guaranteeing that a function is determined by its derivatives at a point. These questions remain active in PDE regularity theory and the study of divergent asymptotic series, where optimal truncation ("superasymptotics") squeezes useful accuracy out of series that never converge.