Taylor & Maclaurin Series

A Taylor polynomial $T_N(x)$ is finite — a polynomial of degree $\leq N$. It always exists for any $N$ when $f$ is $N$-times differentiable, and it's an approximation: $f(x) = T_N(x) + R_N(x)$, with the remainder $R_N$ measurable but generally nonzero. The Taylor series is the infinite sum $\sum_{k=0}^\infty f^{(k)}(a)(x-a)^k/k!$ — it's a limit object that may or may not converge, and even if it converges, may not equal $f$ (see the bump-function example). So: polynomial = approximation tool with explicit error, series = formal infinite object that's only useful when it converges to $f$.

It animates how each new term $\dfrac{f^{(k)}(a)}{k!}(x-a)^k$ contributes to the partial sum. Mode shows $T_0 = f(a)$ (a horizontal line), then $T_1$ adds the tangent slope (linear approximation), $T_2$ adds the parabola correction, $T_3$ adds the cubic, and so on — each term highlights in a different color and its contribution is plotted alongside. You can clearly see the Taylor polynomial "wrap" further around $f$: $T_2$ matches function value + slope + curvature at $a$; $T_3$ matches up to the third derivative, and so on. By order $N$, $T_N$ matches all derivatives up to order $N$ at $a$ exactly.

Almost everywhere a non-polynomial function meets a computer. CPUs compute $\sin, \cos, \exp, \log$ via low-order polynomial approximations (often Chebyshev-modified Taylor) — your hardware multiplies and adds, that's it. Physics: small-angle approximation $\sin\theta \approx \theta$ is $T_1$; pendulum equation linearization for SHM. Numerical methods: Newton's method's quadratic convergence comes from Taylor expansion. ODE solvers: Runge-Kutta methods are Taylor-series-matched at intermediate points. Machine learning: gradient descent uses $T_1$, Newton uses $T_2$ (the Hessian). Statistics: delta method approximates variance of $g(X)$ via $T_1$ around $\E[X]$. Quantum mechanics: time-evolution operator $e^{-iHt/\hbar}$ expanded in $t$.

The radius of convergence is determined by the nearest singularity in the complex plane, not just the real line. $1/(1-x)$ has a real singularity at $x=1$ — distance 1 from origin, so $R=1$. $1/(1+x^2)$ has no real singularities, but factoring over $\C$: $1+x^2 = (1+ix)(1-ix)$ — singularities at $x=\pm i$, complex distance from 0 is exactly 1. So the Maclaurin series $1 - x^2 + x^4 - x^6 + \cdots$ converges only for $\abs{x}<1$. This is Carl Friedrich Gauss's deep insight: Taylor series "see" complex singularities even when computed only with real values. This is why $1/(1+x^2)$ misbehaves at $x=\pm 1$ even though $f$ is smooth there.

Use the Lagrange bound: $\abs{R_N(x)} \le \dfrac{1}{(N+1)!}\abs{x}^{N+1}$ since $\abs{\sin^{(k)}}\le 1$. For $\sin(\pi/4)$: need $(\pi/4)^{N+1}/(N+1)! < \eps$. For $\eps = 10^{-6}$: $N=7$ suffices — and because $\sin$'s degree-8 term vanishes, $T_7 = T_8$, so the sharper bound is $(\pi/4)^9/9! \approx 3\times 10^{-7}$ (note $(\pi/4)^8/8! \approx 3.6\times 10^{-6}$ alone would not clear $10^{-6}$). For $\eps = 10^{-15}$ (machine precision): $N=15$. Practical CPUs use a smarter trick: range reduction (use $\sin(x+2\pi k) = \sin(x)$ to bring $x$ to $[-\pi/4, \pi/4]$), then a low-degree minimax polynomial — typically degree 7 or 11 — chosen by Remez algorithm to minimize maximum error rather than error at one point. So Taylor itself is rarely used in CPUs; its idea is.

It's profound. In real analysis, "smooth" (infinitely differentiable) functions can have Taylor series that don't converge, or converge to the wrong function ($e^{-1/x^2}$ is smooth at 0 with all derivatives = 0, but is nonzero elsewhere). In complex analysis, the situation is much cleaner: a function holomorphic on a disk is automatically equal to its Taylor series there. This is Cauchy's integral theorem + Cauchy's formula: $f^{(n)}(a) = \dfrac{n!}{2\pi i}\oint \dfrac{f(z)}{(z-a)^{n+1}}\,dz$. The radius of convergence is exactly the distance to the nearest singularity in $\C$. Functions that equal their Taylor series everywhere are called analytic (or "real-analytic" on $\R$). Holomorphic ⇒ analytic in $\C$, but not in $\R$ — that's why complex analysis is so much "nicer".

Because of the $(x-a)^{N+1}$ factor in the remainder: $\abs{R_N(x)} \le M\abs{x-a}^{N+1}/(N+1)!$. When $\abs{x-a}$ is small, this is tiny — even for low $N$. When $\abs{x-a}$ is moderate, you need large $N$ to overcome the $(x-a)^{N+1}$ factor. When $\abs{x-a}$ exceeds the radius of convergence, no $N$ helps: the remainder grows. Geometrically: $T_N$ matches all derivatives at $a$, so it agrees with $f$ "infinitely closely" at $a$ — but it has no information about $f$'s behavior far from $a$. The polynomial extrapolates based purely on what it sees near $a$, which is only locally meaningful.