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Statement of the theorem

The precise statement of the most basic version of Taylor's theorem is as follows:

Taylor's theorem — Let k ≥ 1 be an integer and let the function f : R → R be k times differentiable at the point a ∈ R. Then there exists a function h : R → R such that

f (x) = f (a) + f ′ (a) (x − a) + f ″ (a) 2 ! (x − a) 2 + ⋯ + f (k) (a) k ! (x − a) k + h k (x) (x − a) k, {\displaystyle f(x)=f(a)+f'(a)(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+\cdots +{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}+h_{k}(x)(x-a)^{k},} and lim x → a h k (x) = 0. {\displaystyle \lim _{x\to a}h_{k}(x)=0.} This is called the Peano form of the remainder.

The polynomial appearing in Taylor's theorem is the k {\textstyle {\boldsymbol {k}}} -th order Taylor polynomial

P k (x) = f (a) + f ′ (a) (x − a) + f ″ (a) 2 ! (x − a) 2 + ⋯ + f (k) (a) k ! (x − a) k {\displaystyle P_{k}(x)=f(a)+f'(a)(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+\cdots +{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}}

of the function f at the point a. The Taylor polynomial is the unique "asymptotic best fit" polynomial in the sense that if there exists a function h : R → R and a k {\textstyle k} -th order polynomial p such that

f (x) = p (x) + h k (x) (x − a) k, lim x → a h k (x) = 0, {\displaystyle f(x)=p(x)+h_{k}(x)(x-a)^{k},\quad \lim _{x\to a}h_{k}(x)=0,}

then p = P. Taylor's theorem describes the asymptotic behavior of the remainder term

R k (x) = f (x) − P k (x), {\displaystyle R_{k}(x)=f(x)-P_{k}(x),}

which is the approximation error when approximating f with its Taylor polynomial. Using the little-o notation, the statement in Taylor's theorem reads as

R k (x) = o (| x − a | k), x → a. {\displaystyle R_{k}(x)=o(|x-a|^{k}),\quad x\to a.}