Taylor Series Expansion

⚡ Key Takeaways

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 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.

DefinitionTaylor Series and Taylor Polynomial

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

1

\quad T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k.

Derivation hint: 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

2

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

Derivation hint: 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:

3

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

Derivation hint: 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.

TheoremTaylor's Theorem with Remainder

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

4

\quad f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k + R_n(x),

Derivation hint: this is an exact equality, not an approximation; $R_n$ absorbs every error. where the Lagrange form of the remainder is

5

\quad R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}

Derivation hint: apply the generalized Mean Value Theorem (Cauchy) repeatedly, or use Rolle's theorem on an auxiliary function; $\xi$ lies strictly between $a$ and $x$ . for some $\xi$ between $a$ and $x$ . The integral form is

6

\quad R_n(x) = \frac{1}{n!}\int_a^x (x-t)^n f^{(n+1)}(t)\,dt.

Derivation hint: integrate by parts $n$ times starting from the Fundamental Theorem of Calculus $f(x)-f(a)=\int_a^x f'(t)\,dt$ . The Taylor series of $f$ converges to $f(x)$ at a point $x$ if and only if $R_n(x)\to 0$ as $n\to\infty$ .

ProofTaylor'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,

7

\quad f(x) = f(a) + \int_a^x f'(t)\,dt.

Derivation hint: this is the $n=0$ base case; the integral is exactly $R_0(x)$ .

Step 2. Integrate $\int_a^x f'(t)\,dt$ by parts choosing $u=f'(t)$ and $dv=dt$ , but use the antiderivative $v=-(x-t)$ (legal since $x$ is constant in the integration):

8

\quad \int_a^x f'(t)\,dt = f'(a)(x-a) + \int_a^x (x-t)f''(t)\,dt.

Derivation hint: $\big[-(x-t)f'(t)\big]_a^x = (x-a)f'(a)$ 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,

9

\quad f(x) = \sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k + \frac{1}{n!}\int_a^x (x-t)^n f^{(n+1)}(t)\,dt.

Derivation hint: 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 $\xi$ between $a$ and $x$ with

10

\quad \frac{1}{n!}\int_a^x (x-t)^n f^{(n+1)}(t)\,dt = \frac{f^{(n+1)}(\xi)}{n!}\int_a^x (x-t)^n\,dt = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}.

Derivation hint: $(x-t)^n$ has constant sign on the interval, so the weighted MVT pulls $f^{(n+1)}$ out at some interior point $\xi$ . $\blacksquare$

∎

mathematician · 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.

🧠 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 $\tfrac{1}{2}f''(a)(x-a)^2$ 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.

Figure

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

ExampleApproximating 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

11

\quad e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}+\cdots.

Derivation hint: substitute $f^{(n)}(0)=1$ into the Maclaurin formula [3].

Step 2 — evaluate $T_3$ at $x=0.1$ .

12

\quad T_3(0.1) = 1 + 0.1 + \frac{0.01}{2} + \frac{0.001}{6} = 1.1051\overline{6}.

Derivation hint: plug $x=0.1$ into [11] truncated at $n=3$ .

Step 3 — bound the error. By Lagrange [5] with $n=3$ , $\xi\in(0,0.1)$ :

13

\quad |R_3| = \frac{e^{\xi}}{4!}(0.1)^4 \le \frac{e^{0.1}}{24}(10^{-4}) < \frac{1.2}{24}\times 10^{-4} = 5\times 10^{-6}.

Derivation hint: bound $e^\xi$ by $e^{0.1}<1.2$ on the interval.

Answer: $\boxed{e^{0.1} \approx 1.105167,\ \text{error} < 5\times10^{-6}}$ (dimensionless). The true value is $1.1051709\ldots$ , confirming the bound.

ExampleCurvature of a cosine and a small-angle limit

Given: A pendulum's restoring quantity involves $\cos\theta$ . Approximate $\cos\theta$ for small $\theta$ (radians) and evaluate at $\theta = 0.2$ rad.

Step 1 — Maclaurin series of cosine. Derivatives cycle $\cos,-\sin,-\cos,\sin$ , giving values $1,0,-1,0$ at $0$ :

14

\quad \cos\theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots.

Derivation hint: only even derivatives survive; alternating signs from the second-derivative sign flips.

Step 2 — second-order approximation. Keep through $\theta^2$ :

15

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

Derivation hint: truncate [14] after the quadratic term.

Step 3 — error from the next term. The leading omitted term is $\theta^4/24 = (0.2)^4/24 = 6.67\times10^{-5}$ .

Answer: $\boxed{\cos(0.2)\approx 0.98,\ \text{error}\approx 6.7\times10^{-5}}$ (dimensionless). The small-angle approximation $\cos\theta\approx 1-\theta^2/2$ is now justified quantitatively.

⚠️ Common mistake

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.

⚠️ Common mistake

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.

CounterexampleCauchy's flat function

Define

16

\quad f(x) = \begin{cases} e^{-1/x^2}, & x\neq 0\\ 0, & x=0.\end{cases}

Derivation hint: 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 $\ne$ analyticity.

connection

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.

🔬 Where this appears

⏳ History

- 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

\ne

analyticity.

- 1840s — Cauchy and Weierstrass develop complex power series, explaining real radii of convergence via complex singularities.

open-problem

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.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also