Equations · equation
Taylor Series
Calculus · 1715
- Equation
- f(x) = Σ_{n=0}^∞ f⁽ⁿ⁾(a) (x − a)ⁿ / n!
- Field
- Calculus
- Named After
- Brook Taylor
- Symbols
- f a smooth function · f⁽ⁿ⁾(a) its nth derivative at the point a · n! n factorial
- Year
- 1715
Significance
Any smooth function, rebuilt near a point from its derivatives there: a constant, plus a slope, plus a curvature, and so on. Keep one term and you have linear approximation; keep them all and e^x, sin x and cos x become polynomials — which is how Euler's identity is proved. It is the machinery behind every 'for small x' in physics, from the pendulum to the low-speed limit of relativity.
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