What is a Taylor series?

A Taylor series represents a smooth function as an infinite sum of polynomial terms, each based on the function's derivatives at a single point. When centered at x=0, it is called a Maclaurin series.

What is the Taylor series for sin(x)?

The Taylor series for sin(x) centered at 0 is: x - x³/3! + x⁵/5! - x⁷/7! + ... Each term adds an odd power of x divided by its factorial, with alternating signs. Just 7 terms gives excellent accuracy from -π to π.

What is the radius of convergence?

The radius of convergence is the range of x values where the Taylor series converges to the actual function. For sin(x), cos(x), and e^x it is infinite. For ln(1+x) it is |x| < 1, and for 1/(1-x) it is also |x| < 1.

Taylor Series Builder

Add polynomial terms one by one and watch the approximation get closer to the real function. The pink shading shows the error.

Controls

Function

sin(x) cos(x) eˣ ln(1+x) 1/(1-x)

Terms (3)

Show error shading

Polynomial

x - x³/6

The Math Behind It

Taylor Series Formula

f(x) = Σ f⁽ⁿ⁾(a)/n! · (x-a)ⁿ
Centered at a=0 (Maclaurin series)
Each term adds one more derivative's worth of information
More terms = better approximation near the center

Common Series

sin(x) = x - x³/3! + x⁵/5! - ...
cos(x) = 1 - x²/2! + x⁴/4! - ...
eˣ = 1 + x + x²/2! + x³/3! + ...

Try This

Start with 1 term for sin(x) — it's just a straight line (y=x)
Add to 3 terms — suddenly it curves like sin!
At 7 terms, it's nearly perfect from -π to π
Try eˣ — notice it converges everywhere, not just near 0
Try ln(1+x) — it only converges for |x| < 1 (radius of convergence)

Key Insight

Taylor series show that smooth functions are secretly polynomials in disguise — you just need infinitely many terms.

Taylor Series Builder

Controls

Polynomial

The Math Behind It

Taylor Series Formula

Common Series

Try This

Key Insight

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Derivative Visualizer

Limits & Continuity

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