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Taylor Series

★★★★★Graduate+

📖Definition

A series expansion that represents a function as an infinite polynomial.

f(x) = ∑_n=0^∈fty \fracf^(n)(a)n!(x-a)ⁿ

Taylor series about x=a

e^x = ∑_n=0^∈fty (xⁿ)/(n!) = 1 + x + (x²)/(2!) + (x³)/(3!) + ...

Taylor series for e^x

sin x = ∑_n=0^∈fty \frac(-1)ⁿ x^2n+1(2n+1)!

Taylor series for sine

cos x = ∑_n=0^∈fty \frac(-1)ⁿ x^2n(2n)!

Taylor series for cosine

예제 1★★★★☆

Find the Taylor polynomial of degree 4 for e^x about x=0.

Discovered by: Brook Taylor (1715)

Colin Maclaurin studied the special case at x=0.

Polynomial approximation of functions

Linearization, perturbation theory

Trig/exponential function calculation

#analysis#series#taylor#advanced