Taylor Polynomials, Remainder Theorems & Applied Expansions

1. Motivation: Polynomial Interpolation of Differential Jets

Let $f: I \to \mathbb{R}$ be $n$ times continuously differentiable on an open interval $I$ containing $x_0$. The $n$-th degree Taylor polynomial $P_n(x)$ centered at $x_0$ is the unique polynomial of degree at most $n$ whose derivatives at $x_0$ match those of $f$ up to order $n$:

When centered at $x_0 = 0$, $P_n(x)$ is specifically called the Maclaurin polynomial.

2. Standard Maclaurin Expansions

• $e^x = \sum_{k=0}^\infty \frac{x^k}{k!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \quad (R = \infty)$
• $\sin(x) = \sum_{k=0}^\infty \frac{(-1)^k x^{2k+1}}{(2k+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots \quad (R = \infty)$
• $\cos(x) = \sum_{k=0}^\infty \frac{(-1)^k x^{2k}}{(2k)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots \quad (R = \infty)$
• $\frac{1}{1 - x} = \sum_{k=0}^\infty x^k = 1 + x + x^2 + x^3 + \dots \quad (R = 1)$
• $\ln(1 + x) = \sum_{k=1}^\infty \frac{(-1)^{k-1} x^k}{k} = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \quad (R = 1)$
• $(1 + x)^\alpha = \sum_{k=0}^\infty \binom{\alpha}{k} x^k = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!} x^2 + \dots \quad (R = 1)$

1. Statement of Taylor's Theorem

Let $f \in C^{n+1}([a, b])$ with $x_0, x \in [a, b]$. Then $f(x) = P_n(x) + R_n(x)$, where $R_n(x)$ is the remainder (truncation error).

2. Lagrange Form of the Remainder

for some intermediate point $c$ strictly between $x_0$ and $x$. Notice that for $n = 0$, this is identically the Lagrange Mean Value Theorem $f(x) - f(x_0) = f'(c)(x - x_0)$.

3. Integral Form of the Remainder

Repeated integration by parts establishes:

4. The Lagrange Truncation Error Bound

If $|f^{(n+1)}(t)| \le M$ for all $t$ between $x_0$ and $x$:

1. Error Budgeting Principle

To approximate $f(x)$ with error strictly less than a specified tolerance $\varepsilon > 0$, we find the smallest integer $n$ such that:

2. High-Precision Approximation of $e$

Expanding $e^x$ at $x_0 = 0$ for $x = 1$ with $M = \max_{c \in [0, 1]} e^c = e < 3$:

For $\varepsilon = 10^{-6}$, setting $\frac{3}{(n+1)!} < 10^{-6} \implies (n+1)! > 3 \times 10^6 \implies n = 9$ (since $10! = 3,628,800$). Thus, adding just 10 terms of $P_9(1)$ guarantees 6 decimal places of accuracy!

1. Integrating Non-Elementary Integrands

Many foundational integrals in physics and probability (such as the error function $\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt$ and the sine integral $\operatorname{Si}(x) = \int_0^x \frac{\sin t}{t} dt$) have no closed-form elementary antiderivative. Taylor series integration resolves them completely.

Expanding $e^{-t^2} = \sum_{k=0}^\infty \frac{(-1)^k t^{2k}}{k!}$:

Because this is an alternating series, the truncation error after $N$ terms is bounded by the magnitude of the $(N+1)$-th term.

1. Physics: Relativistic Kinetic Energy Correction

Einstein's relativistic total energy is $E = \gamma m c^2$ with $\gamma = (1 - v^2/c^2)^{-1/2}$. Expanding via the binomial Taylor series for $x = v^2/c^2 \ll 1$:

The kinetic energy $K = E - mc^2 = (\gamma - 1)mc^2$ becomes:

The leading Taylor term recovers Newtonian kinetic energy $\frac{1}{2}mv^2$, while the second term provides the leading relativistic correction!

2. Economics: Arrow-Pratt Risk Aversion Measure

Expanding an agent's expected utility $U(w + \tilde{z})$ around wealth $w$ reveals that risk premium is proportional to $-\frac{U''(w)}{U'(w)}$.

3. Biology: Population Linearization

Expanding the non-linear logistic growth equation $\frac{dN}{dt} = r N (1 - N/K)$ around the carrying capacity $K$ yields stable exponential decay of perturbations.