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.

Taylor's Theorem Example

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].
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