Math basics › Calculus II
Taylor series
The idea
A polynomial that matches a function's value, slope, bend and so on at one point. More terms fit it better near that point.
The formula
f(x) = Σ f⁽ⁿ⁾(a)/n! · (x − a)ⁿ. Centered at a = 0 it is called a Maclaurin series.
Ones worth knowing
eˣ = 1 + x + x²/2! + x³/3! + …, sin x = x − x³/3! + x⁵/5! − …, and 1/(1 − x) = 1 + x + x² + … when |x| < 1.
Where it works
A power series converges on an interval around its center. Find the radius with the ratio test, then check each endpoint on its own.