Taylor's theorem proof, example

Taylor's theorem shows the approximation of n times differentiable function around a given point by an n-th order Taylor-polynomial.

Proof:

If n ≥ 0 and f(x) is (n+1)-times differentiable on the open interval (a, x) and n times differntiable on the closed interval (a,x), then

n! denotes the factorial of n and Rn(x) is the remainder term.

The remainder term is the difference between the Taylor polynomial of degree(n) and the original function.

where ξ is some number from the interval [a, x].

Taylor's theorem proof, example

Taylor's theorem shows the approximation of n times differentiable function around a given point by an n-th order Taylor-polynomial.

calculators

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2024-10-15

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