Calculus

Taylor Series: Polynomial Clones

What if you could replace a complicated function like sin(x) or e^x with a simple polynomial that behaves almost the same way? That is exactly what Taylor series do.

The idea: match a function’s value, slope, curvature, and higher derivatives at a single point, one term at a time.

1. The Taylor Polynomial Formula

The Taylor series of f(x) centered at a is:

f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots

Each term matches one more derivative of f at the center point. The more terms you include, the better the polynomial mimics the original function.

2. Approximating sin(x)

The Taylor series for sin(x) centered at 0 is:

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

Use the slider to add more terms and watch the polynomial wrap itself around the sine curve.

number of terms1

110

Try This

Try this: Start with 1 term (just “x”) — it only matches near the origin. Add term 2 and the polynomial bends to follow sin(x) further. By 5 terms, the match is excellent from about -6 to 6. The polynomial “grows outward” from the center, conquering more territory with each new term.

3. Approximating e^x

The exponential function has the simplest Taylor series of all:

e^x \approx 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots

Every derivative of e^x is e^x, so every coefficient is 1/n!.

number of terms1

110

Connection

For positive x, the polynomial always undershoots e^x (each new term adds more positive value). For negative x, the polynomial oscillates above and below, converging from both sides. With enough terms, the match is excellent across a wide range.

4. Approximating cos(x)

\cos(x) \approx 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots

number of terms1

5. Changing the Center Point

Taylor series do not have to be centered at 0. Centering at a different point a makes the approximation best near a. Let’s see this with ln(x) centered at a:

\ln(x) \approx \ln(a) + \frac00(x-a) - \frac0{2a^2}(x-a)^2 + \frac0{3a^3}(x-a)^3

a (center point)1

0.55

Try This

Try this: Set the center to a = 1 and the polynomial matches ln(x) nicely near x = 1. Move the center to a = 3 and the polynomial shifts to match near x = 3 instead. The approximation is always best near the center point and gets worse as you move away.

6. Why Taylor Series Matter

Taylor series are not just a mathematical trick. They are how calculators compute sin, cos, e^x, and logarithms. Your phone evaluates a polynomial — fast, simple arithmetic — rather than computing the “true” transcendental function.

They also reveal deep connections. For example, the Taylor series for e^(ix) combines the series for cos(x) and sin(x), leading to Euler’s famous formula:

e^0 = \cos(x) + i\sin(x)

Challenge

Challenge: Write out the first 4 nonzero terms of the Taylor series for e^x, then substitute ix for x. Group the real and imaginary parts. You should see the Taylor series for cos(x) and i*sin(x) appear separately. This is one of the most beautiful results in all of mathematics.

The Big Idea

A Taylor series replaces any smooth function with a polynomial that matches it perfectly near a chosen center point. More terms mean a better match over a wider range.

Polynomials are the simplest functions to evaluate — just multiply and add. Taylor series let you trade any complicated function for a polynomial clone, as accurate as you need. It is approximation elevated to an exact science.

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