Taylor Series and Approximation

Turning Functions into Polynomials

Computers cannot directly evaluate $\sin x$, $e^x$, or $\ln x$ — these are defined by limits, not arithmetic. The resolution is Taylor series: any smooth function can be approximated (often exactly represented) by a polynomial built from its derivatives at a single point. Since polynomials need only addition and multiplication, this is how calculators and libraries actually compute these functions.

The Taylor series of $f$ about a point $a$ is $$f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \dots$$ Each term uses a higher derivative and a higher power. Intuitively: the constant term matches the function's value, the linear term matches its slope, the quadratic term matches its curvature, and so on — each term correcting the approximation a little more. Truncating after a few terms gives a Taylor polynomial that hugs the function near $a$.

The special case $a = 0$ (a Maclaurin series) gives famously clean expansions: $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$$ and similar series for $\sin x$ and $\cos x$. Keeping more terms improves accuracy; the linear approximation $f(x) \approx f(a) + f'(a)(x-a)$ (just the first two terms) is the simplest and is the basis of Newton's method and many numerical algorithms.

The key idea for computing: replace a hard function by an easy polynomial that matches it locally. The approximation is best near the expansion point $a$ and generally degrades as you move away — so choosing a good center, and enough terms, is a practical trade-off between accuracy and cost.

Common pitfall: expecting a truncated Taylor polynomial to be accurate far from the expansion point. A Taylor polynomial matches the function best near the center $a$ and its error typically grows as $|x - a|$ increases — a few terms good near $a$ can be wildly wrong far away. More terms extend the accurate region but never make a finite polynomial match everywhere; the approximation is fundamentally local.