Editing Taylor's theorem (section)

=== Statement of the theorem ===

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

{{math theorem
| name = Taylor's theorem<ref>{{ citation|first1=Angelo|last1=Genocchi|first2= Giuseppe|last2=Peano|title=Calcolo differenziale e principii di calcolo integrale|location=(N. 67, pp.&nbsp;XVII–XIX)|publisher=[[Fratelli Bocca |Fratelli Bocca ed.]]|year=1884}}</ref><ref>{{Citation | last1=Spivak | first1=Michael | author1-link=Michael Spivak | title=Calculus  | publisher=Publish or Perish | location=Houston, TX | edition=3rd | isbn=978-0-914098-89-8 | year=1994| page=383}}</ref><ref>{{springer|title=Taylor formula|id=p/t092300}}</ref>
| math_statement = Let ''k''&nbsp;≥&nbsp;1 be an [[integer]] and let the [[Function (mathematics)|function]] {{nowrap|''f'' : '''R''' → '''R'''}} be ''k'' times [[Differentiable function|differentiable]] at the point {{nowrap|''a'' ∈ '''R'''}}. Then there exists a function  {{nowrap|''h<sub>k</sub>'' : '''R''' → '''R'''}} such that

<math display="block"> f(x) = \sum_{i=0}^k \frac{f^{(i)}(a)}{i!}(x-a)^i + h_k(x)(x-a)^k,</math>
and
<math display="block">\lim_{x\to a} h_k(x) = 0.</math>
This is called the '''[[Peano]] form of the remainder'''.
}}

The polynomial appearing in Taylor's theorem is the '''<math display="inline">\boldsymbol{k}</math>-th order Taylor polynomial'''

<math display="block">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 </math>

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  {{nowrap|''h<sub>k</sub>'' : '''R''' → '''R'''}} and a <math display="inline">k</math>-th order polynomial ''p'' such that

<math display="block"> f(x) = p(x) + h_k(x)(x-a)^k, \quad \lim_{x\to a} h_k(x) = 0 ,</math>

then ''p''&nbsp;=&nbsp;''P<sub>k</sub>''. Taylor's theorem describes the asymptotic behavior of the '''remainder term'''

<math display="block"> R_k(x) = f(x) - P_k(x),</math>

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

<math display="block">R_k(x) = o(|x-a|^{k}), \quad x\to a.</math>