Skip to content
Formal Sciences
Mathematics
1715
Intermediate

Taylor Series

f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

Any smooth function can be approximated as an infinite sum of polynomial terms around a point.

By Brook Taylor, James Gregory

Formal Sciences
Taylor Series
1715 · Brook Taylor
Human Reviewed
84%

Rabbit Hole Mode

Five doors into the universe behind this equation. Choose your path.

Story PortalWho discovered this?Discover how Brook Taylor and others shaped this equation.Visual PortalWhat does it look like?See the equation come alive in the Visual Studio.Machine PortalWhere is it used?Explore machines powered by this equation — numerical computing.Math PortalWhat does it derive from?Trace the mathematical lineage from eulers identity.Future PortalWhere is it going?Neural Taylor expansions
Launch collection — full story, visual, sound & share card
Why it matters: Made calculus computable—engineers could approximate any function with polynomials.

Discoverers: Brook Taylor, James Gregory (1715)

What does it mean?

Any smooth function can be approximated as an infinite sum of polynomial terms around a point.

Why should I care?

Made calculus computable—engineers could approximate any function with polynomials.

Equation Compass

North — Prerequisites

West — History

East — Applications

South — Derivations

Variables & Units

SymbolNameUnitMeaning
f(x)f(x)Function—Function to expand
aaCenter—Expansion point
nnOrder—Term index

Worked Example

e^x ≈ 1 + x + x²/2! + x³/3! + ... near x = 0.

AI Guide (Pro)

Ask questions about equations and get answers grounded in the Equation Universe catalog.

Upgrade to ProTry demo (7 days)

Sources & further reading

Search WikipediaSearch Wolfram MathWorldOpen in Wolfram|AlphaWatch explainers on YouTube

Share this equation

Equation Universe

Taylor Series

f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

Real-world impact

Intelligent systems

Mathematics trains models that reshape work and creativity.

Photo: Unsplash — AI concept

Any smooth function can be approximated as an infinite sum of polynomial terms around a point.

equation-universe.vercel.app

Post