Taylor Series

The problem with transcendental functions#

You know how to evaluate $x^2$ or $x^3 + 4x$ for any number. Polynomials are just arithmetic — multiplication and addition. But $\sin(x)$, $\cos(x)$, and $e^x$ are different beasts. They're transcendental functions: no finite combination of basic arithmetic produces them exactly.

So when a calculator displays $\sin(1.5) = 0.9975$, it's not doing something magical. It's doing arithmetic. It's using a polynomial that looks so much like $\sin(x)$ near that point that the difference is smaller than any digit it could show you.

The machinery behind this is the Taylor series — one of the most powerful ideas in mathematics.

A polynomial that matches perfectly at one point#

Start with a simple question: can I find a polynomial $P(x)$ that behaves exactly like $f(x) = \sin(x)$ at $x = 0$?

"Exactly like" means matching not just the value, but the slope, the curvature, the rate of change of curvature — every derivative, forever. This is surprisingly achievable. The Taylor series around $x = 0$ (also called the Maclaurin series) is:

$$f(x) = f(0) + f'(0)\,x + \frac{f''(0)}{2!}\,x^2 + \frac{f'''(0)}{3!}\,x^3 + \cdots = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}\,x^n$$

Each term captures one more layer of how $f$ behaves at 0. Adding more terms doesn't just make the approximation better at 0 — it extends accurate behavior further and further from 0.

Sine, cosine, and eˣ#

For $\sin(x)$, the derivatives cycle: $\sin$, $\cos$, $-\sin$, $-\cos$, and repeat. At $x = 0$, $\sin(0) = 0$, $\cos(0) = 1$, so every other coefficient vanishes:

$$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$

For $e^x$, every derivative of $e^x$ is $e^x$. At $x = 0$, every derivative equals 1:

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$

Watch what happens when you add terms one by one — the polynomial starts matching the true function near 0 and gradually extends its accuracy outward.

Notice how $e^x$ behaves differently from $\sin(x)$. The exponential grows unboundedly, so the polynomial needs more terms to keep up at larger $x$. The sine, bounded between −1 and 1, is easier — ten terms give nearly perfect accuracy across many full cycles.

Why factorials in the denominator?#

The $n!$ in the denominator isn't an arbitrary choice — it's forced by the requirement that derivatives match. To ensure the $n$-th derivative of our polynomial $P(x)$ matches $f^{(n)}(0)$, we need:

$$P(x) = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots$$

Taking the $n$-th derivative of $c_n x^n$ gives $c_n \cdot n!$. Setting that equal to $f^{(n)}(0)$ gives $c_n = f^{(n)}(0) / n!$.

The factorial is the natural normalizer that falls out of repeated differentiation.

Convergence and radius#

Not every Taylor series converges everywhere. The series for $\sin(x)$ and $e^x$ converge for all real numbers. But the series for $\ln(1 + x)$:

$$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots$$

only converges for $-1 < x \leq 1$. Outside this radius of convergence, the partial sums fly off to infinity instead of settling. The complex analysis reason is that $\ln(1+x)$ has a singularity at $x = -1$, and a circle of convergence centered at 0 can only extend as far as that nearest singularity.

Add terms and watch the split personality of the partial sums: inside the shaded interval $(-1, 1]$ the polynomial hugs $\ln(1+x)$ ever more tightly, but cross either edge of the radius and more terms make things worse, the curve swinging off violently. More terms only ever help inside the radius of convergence — beyond it, the series is simply the wrong tool.

Practical power: fast computation#

The reason Taylor series matter to every programmer, physicist, and engineer is that they turn transcendental functions into arithmetic. A modern CPU can't compute $\sin(x)$ natively — but it can multiply and add numbers incredibly fast. The standard library implementation of sin() in C is essentially a carefully chosen polynomial (often using a variant of CORDIC or a minimax polynomial) accurate to 64-bit precision.

The same idea underlies numerical methods, signal processing, and differential equation solvers. Whenever you approximate a smooth function near a known point, you're implicitly using Taylor's idea — that smoothness creates a polynomial fingerprint.