View source for Taylor's theorem

{{short description|Approximation of a function by a truncated power series}}
[[File:Taylorspolynomialexbig.svg|thumb|right|300px|The exponential function <math display="inline">y=e^x</math> (red) and the corresponding Taylor polynomial of degree four (dashed green) around the origin.]]
{{Calculus |Differential}}

In [[calculus]], '''Taylor's theorem''' gives an approximation of a <math display="inline">k</math>-times [[differentiable function]] around a given point by a [[polynomial]] of degree <math display="inline">k</math>, called the <math display="inline">k</math>-th-order '''Taylor polynomial'''. For a [[smooth function]], the Taylor polynomial is the truncation at the order ''<math display="inline">k</math>'' of the [[Taylor series]] of the function. The first-order Taylor polynomial is the [[linear approximation]] of the function, and the second-order Taylor polynomial is often referred to as the '''quadratic approximation'''.<ref>(2013). [http://www.math.ubc.ca/~sujatha/2013/103/week10-12/Linearapp.pdf"Linear and quadratic approximation"] Retrieved December 6, 2018</ref> There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.

Taylor's theorem is named after the mathematician [[Brook Taylor]], who stated a version of it in 1715,<ref>{{cite book|language=la|last=Taylor |first=Brook |title=Methodus Incrementorum Directa et Inversa |url=https://archive.org/details/UFIE003454_TO0324_PNI-2529_000000|trans-title=Direct and Reverse Methods of Incrementation |location=London |date=1715 |at=p. 21–23 (Prop. VII, Thm. 3, Cor. 2)}} Translated into English in {{cite book|first=D. J. |last=Struik|title=A Source Book in Mathematics 1200–1800 |location=Cambridge, Massachusetts |publisher=Harvard University Press |date=1969 |pages= 329–332}}</ref> although an earlier version of the result was already mentioned in [[1671 in science|1671]] by [[James Gregory (astronomer and mathematician)|James Gregory]].<ref>{{harvnb|Kline|1972|pp=442, 464}}.</ref>

Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in [[mathematical analysis]]. It gives simple arithmetic formulas to accurately compute values of many [[transcendental function]]s such as the [[exponential function]] and [[trigonometric function]]s.
It is the starting point of the study of [[analytic function]]s, and is fundamental in various areas of mathematics, as well as in [[numerical analysis]] and [[mathematical physics]]. Taylor's theorem also generalizes to [[multivariate function|multivariate]] and [[vector valued function|vector valued]] functions. It provided the mathematical basis for some landmark early computing machines: [[Charles Babbage]]'s [[Difference Engine]] calculated sines, cosines, logarithms, and other transcendental functions by numerically integrating the first 7 terms of their Taylor series.

== Motivation ==
[[File:E^x with linear approximation.png|thumb|right|Graph of <math display="inline">f(x)=e^x</math> (blue) with its [[linear approximation]] <math display="inline">P_1(x)=1+x</math>  (red) at <math display="inline">a=0</math>.]]
If a real-valued [[function (mathematics)|function]] <math display="inline">f(x)</math> is [[Derivative|differentiable]] at the point <math display="inline">x=a</math>, then it has a [[linear approximation]] near this point.  This means that there exists a function ''h''<sub>1</sub>(''x'') such that

<math display="block"> f(x) = f(a) + f'(a)(x - a) + h_1(x)(x - a), \quad \lim_{x \to a} h_1(x) = 0.</math>

Here

<math display="block">P_1(x) = f(a) + f'(a)(x - a)</math>

is the linear approximation of <math display="inline">f(x)</math> for ''x'' near the point ''a'', whose graph <math display="inline">y=P_1(x)</math> is the [[tangent line]] to the graph <math display="inline">y=f(x)</math> at {{nowrap|1=''x'' = ''a''}}. The error in the approximation is:
<math display="block">R_1(x) = f(x) - P_1(x) = h_1(x)(x - a).</math>

As ''x'' tends to&nbsp;''a,'' this error goes to zero much faster than <math>f'(a)(x{-}a)</math>, making <math>f(x)\approx P_1(x)</math> a useful approximation.

[[File:E^x with quadratic approximation corrected.png|thumb|right|Graph of <math display="inline">f(x)=e^x</math> (blue) with its quadratic approximation  <math>P_2(x) = 1 +x + \dfrac{x^2}{2}</math> (red) at <math display="inline">a=0</math>. Note the improvement in the approximation.]]
For a better approximation to <math display="inline">f(x)</math>, we can fit a [[quadratic polynomial]] instead of a linear function:

<math display="block">P_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2.</math>

Instead of just matching one derivative of <math display="inline">f(x)</math> at  <math display="inline">x=a</math>, this polynomial has the same first and second derivatives, as is evident upon differentiation.

Taylor's theorem ensures that the ''quadratic approximation'' is, in a sufficiently small neighborhood of <math display="inline">x=a</math>, more accurate than the linear approximation.  Specifically,

<math display="block">f(x) = P_2(x) + h_2(x)(x - a)^2, \quad \lim_{x \to a} h_2(x) = 0.</math>

Here the error in the approximation is

<math display="block">R_2(x) = f(x) - P_2(x) = h_2(x)(x - a)^2,</math>

which, given the limiting behavior of <math>h_2</math>, goes to zero faster than <math>(x - a)^2</math> as ''x'' tends to&nbsp;''a''.

[[File:Tayloranimation.gif|thumb|360px|right|Approximation of <math display="inline">f(x)= \dfrac{1}{1+x^2}</math> (blue) by its Taylor polynomials <math display="inline">P_k</math> of order <math display="inline">k=1,\ldots,16</math> centered at <math display="inline">x=0</math> (red) and <math display="inline">x=1</math> (green). The approximations do not improve at all outside <math>(-1,1)</math> and <math display="inline">(1-\sqrt{2}, 1+\sqrt{2})</math>, respectively.]] Similarly, we might get still better approximations to ''f'' if we use [[polynomial]]s of higher degree, since then we can match even more derivatives with ''f'' at the selected base point.

In general, the error in approximating a function by a polynomial of degree ''k'' will go to zero much faster than <math>(x-a)^k</math> as ''x'' tends to&nbsp;''a''. However, there are functions, even infinitely differentiable ones, for which increasing the degree of the approximating polynomial does not increase the accuracy of approximation: we say such a function fails to be [[Analytic function|analytic]] at ''x = a'': it is not (locally) determined by its derivatives at this point.

Taylor's theorem is of asymptotic nature: it only tells us that the error <math display="inline">R_k</math> in an [[approximation]] by a <math display="inline">k</math>-th order Taylor polynomial ''P<sub>k</sub>'' tends to zero faster than any nonzero <math display="inline">k</math>-th degree [[polynomial]] as <math display="inline">x \to a</math>. It does not tell us how large the error is in any concrete [[neighborhood (mathematics)|neighborhood]] of the center of expansion, but for this purpose there are explicit formulas for the remainder term (given below) which are valid under some additional regularity assumptions on ''f''. These enhanced versions of Taylor's theorem typically lead to [[uniform convergence|uniform estimates]] for the approximation error in a small neighborhood of the center of expansion, but the estimates do not necessarily hold for neighborhoods which are too large, even if the function ''f'' is [[analytic function|analytic]]. In that situation one may have to select several Taylor polynomials with different centers of expansion to have reliable Taylor-approximations of the original function (see animation on the right.)

There are several ways we might use the remainder term:

# Estimate the error for a polynomial ''P<sub>k</sub>''(''x'') of degree ''k'' estimating <math display="inline">f(x)</math> on a given interval (''a'' – ''r'', ''a'' + ''r''). (Given the interval and degree, we find the error.)
# Find the smallest degree ''k'' for which the polynomial ''P<sub>k</sub>''(''x'') approximates <math display="inline">f(x)</math> to within a given error tolerance on a given interval (''a'' − ''r'', ''a'' + ''r'') . (Given the interval and error tolerance, we find the degree.)
# Find the largest interval (''a'' − ''r'', ''a'' + ''r'') on which ''P<sub>k</sub>''(''x'') approximates <math display="inline">f(x)</math> to within a given error tolerance. (Given the degree and error tolerance, we find the interval.)

{{clear}}

== Taylor's theorem in one real variable ==

=== 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>

=== Explicit formulas for the remainder ===

Under stronger regularity assumptions on ''f'' there are several precise formulas for the remainder term ''R<sub>k</sub>'' of the Taylor polynomial, the most common ones being the following.

{{math theorem
| name = Mean-value forms of the remainder
| math_statement = Let {{nowrap|''f'' : '''R''' → '''R'''}}  be ''k''&nbsp;+&nbsp;1 times [[Differentiable function|differentiable]] on the [[open interval]] with ''f''{{i sup|(''k'')}} [[continuous function|continuous]] on the [[closed interval]] between <math display=inline>a</math> and <math display=inline>x</math>.<ref>The hypothesis of ''f''{{i sup|(''k'')}} being [[continuous function|continuous]] on the [[closed interval|''closed'' interval]] between <math display=inline>a</math> and <math display=inline>x</math> is ''not'' redundant. Although ''f'' being ''k''&nbsp;+&nbsp;1 times [[Differentiable function|differentiable]] on the [[open interval]] between <math display=inline>a</math> and <math display=inline>x</math> does imply that ''f''{{i sup|(''k'')}} is [[continuous function|continuous]] on the [[open interval|''open'' interval]] between <math display=inline>a</math> and <math display=inline>x</math>, it does ''not'' imply that ''f''{{i sup|(''k'')}} is [[continuous function|continuous]] on the [[closed interval|''closed'' interval]] between <math display=inline>a</math> and <math display=inline>x</math>, i.e. it does not imply that ''f''{{i sup|(''k'')}} is [[continuous function|continuous]] at the ''endpoints'' of that interval. Consider, for example, the [[Function (mathematics)|function]] {{nowrap|''f'' : [0,1] → '''R'''}} defined to equal <math> \sin(1/x)</math> on <math>(0,1]</math> and with <math>f(0)=0</math>. This is not [[continuous function|continuous]] at ''0'', but is [[continuous function|continuous]] on <math>(0,1)</math>. Moreover, one can show that this [[Function (mathematics)|function]] has an [[antiderivative]]. Therefore that [[antiderivative]] is [[Differentiable function|differentiable]] on <math>(0,1)</math>, its [[derivative]] (the function ''f'') is [[continuous function|continuous]] on the [[open interval|''open'' interval]] <math>(0,1)</math>, but its [[derivative]] ''f'' is ''not'' [[continuous function|continuous]] on the [[closed interval|''closed'' interval]] <math>[0,1]</math>. So the theorem would not apply in this case.</ref> Then

<math display="block"> R_k(x) = \frac{f^{(k+1)}(\xi_L)}{(k+1)!} (x-a)^{k+1} </math>

for some real number <math display="inline">\xi_L</math> between <math display=inline>a</math> and <math display=inline>x</math>. This is the '''[[Joseph Louis Lagrange|Lagrange]] form'''<ref>{{harvnb|Kline|1998|loc=§20.3}}; {{harvnb|Apostol|1967|loc=§7.7}}.</ref> of the remainder.

Similarly,

<math display="block"> R_k(x) = \frac{f^{(k+1)}(\xi_C)}{k!}(x-\xi_C)^k(x-a) </math>

for some real number <math display="inline">\xi_C</math> between <math display=inline>a</math> and <math display=inline>x</math>. This is the '''[[Augustin Louis Cauchy|Cauchy]] form'''<ref>{{harvnb|Apostol|1967|loc=§7.7}}.</ref> of the remainder.

Both can be thought of as specific cases of the following result: Consider <math>p>0</math>

<math display="block"> R_k(x) = \frac{f^{(k+1)}(\xi_S)}{k!}(x-\xi_S)^{k+1-p}\frac{(x-a)^p}{p} </math>
for some real number <math display="inline">\xi_S</math> between <math display=inline>a</math> and <math display=inline>x</math>. This is the '''[[Oskar Schlömilch|Schlömilch]] form''' of the remainder (sometimes called the '''Schlömilch-[[Édouard Roche|Roche]]'''). The choice <math display="inline">p=k+1</math> is the Lagrange form, whilst the choice <math display="inline">p=1</math> is the Cauchy form.
}}

These refinements of Taylor's theorem are usually proved using the [[mean value theorem]], whence the name. Additionally, notice that this is precisely the [[mean value theorem]] when <math display="inline">k=0</math>. Also other similar expressions can be found. For example, if ''G''(''t'') is continuous on the closed interval and differentiable with a non-vanishing derivative on the open interval between <math display="inline">a</math> and <math display="inline">x</math>, then

<math display="block"> R_k(x) = \frac{f^{(k+1)}(\xi)}{k!}(x-\xi)^k \frac{G(x)-G(a)}{G'(\xi)} </math>

for some number <math display="inline">\xi</math> between <math display="inline">a</math> and <math display="inline">x</math>. This version covers the Lagrange and Cauchy forms of the remainder as special cases, and is proved below using [[mean value theorem#Cauchy's mean value theorem|Cauchy's mean value theorem]]. The Lagrange form is obtained by taking <math>G(t)=(x-t)^{k+1}</math> and the Cauchy form is obtained by taking <math>G(t)=t-a</math>.

{{anchor|Integral form of the remainder}}The statement for the integral form of the remainder is more advanced than the previous ones, and requires understanding of [[Lebesgue integral|Lebesgue integration theory]] for the full generality. However, it holds also in the sense of [[Riemann integral]] provided the (''k''&nbsp;+&nbsp;1)th derivative of ''f'' is continuous on the closed interval [''a'',''x''].

{{math theorem|name=Integral form of the remainder<ref>{{harvnb|Apostol|1967|loc=§7.5}}.</ref> |math_statement=Let <math display=inline>f^{(k)}</math> be [[absolutely continuous]] on the [[closed interval]] between <math display=inline>a</math> and <math display=inline>x</math>. Then

<math display="block"> R_k(x) = \int_a^x \frac{f^{(k+1)} (t)}{k!} (x - t)^k \, dt. </math>

}}

Due to the [[absolutely continuous|absolute continuity]] of ''f''{{i sup|(''k'')}} on the [[closed interval]] between <math display=inline>a</math> and <math display=inline>x</math>, its derivative ''f''{{i sup|(''k''+1)}} exists as an ''L''{{i sup|1}}-function, and the result can be [[#Derivation for the integral form of the remainder|proven]] by a formal calculation using the [[fundamental theorem of calculus]] and [[integration by parts]].

=== Estimates for the remainder ===

It is often useful in practice to be able to estimate the remainder term appearing in the Taylor approximation, rather than having an exact formula for it.  Suppose that ''f'' is {{nowrap|(''k'' + 1)}}-times continuously differentiable in an interval ''I'' containing ''a''. Suppose that there are real constants ''q'' and ''Q'' such that

<math display="block">q\le f^{(k+1)}(x)\le Q</math>

throughout ''I''. Then the remainder term satisfies the inequality<ref>{{harvnb|Apostol|1967|loc=§7.6}}</ref>

<math display="block">q\frac{(x-a)^{k+1}}{(k+1)!}\le R_k(x)\le Q\frac{(x-a)^{k+1}}{(k+1)!},</math>

if {{nowrap|''x'' > ''a''}}, and a similar estimate if {{nowrap|''x'' < ''a''}}. This is a simple consequence of the Lagrange form of the remainder. In particular, if

<math display="block">|f^{(k+1)}(x)|\le M</math>

on an interval {{nowrap|1=''I'' = (''a'' − ''r'',''a'' + ''r'')}} with some <math>r > 0</math> , then

<math display="block">|R_k(x)|\le M\frac{|x-a|^{k+1}}{(k+1)!}\le M\frac{r^{k+1}}{(k+1)!}</math>

for all {{nowrap|''x''∈(''a'' − ''r'',''a'' + ''r'').}} The second inequality is called a [[uniform convergence|uniform estimate]], because it holds uniformly for all ''x'' on the interval {{nowrap|(''a'' − ''r'',''a'' + ''r'').}}

=== Example ===

[[File:Expanimation.gif|thumb|400px|right|Approximation of <math display="inline">e^x</math> (blue) by its Taylor polynomials <math>P_k</math> of order <math display="inline">k=1,\ldots,7</math> centered at <math display="inline">x=0</math> (red).]] Suppose that we wish to find the approximate value of the function <math display="inline">f(x)=e^x</math> on the interval <math display="inline">[-1,1]</math> while ensuring that the error in the approximation is no more than 10<sup>−5</sup>. In this example we pretend that we only know the following properties of the exponential function:

{{NumBlk|:|<math>e^0=1, \qquad \frac{d}{dx} e^x = e^x, \qquad e^x>0, \qquad x\in\R.</math>|{{EquationRef|★}}}}

From these properties it follows that <math display="inline">f^{(k)}(x)=e^x</math> for all <math display="inline">k</math>, and in particular, <math display="inline">f^{(k)}(0)=1</math>. Hence the ''<math display="inline">k</math>''-th order Taylor polynomial of <math display="inline">f</math> at <math display="inline">0</math> and its remainder term in the Lagrange form are given by

<math display="block"> P_k(x) = 1+x+\frac{x^2}{2!}+\cdots+\frac{x^k}{k!}, \qquad R_k(x)=\frac{e^\xi}{(k+1)!}x^{k+1},</math>

where <math display="inline">\xi</math> is some number between 0 and ''x''. Since ''e''<sup>''x''</sup> is increasing by ({{EquationNote|★}}), we can simply use <math display="inline">e^x \leq 1</math> for <math display="inline">x \in [-1,0]</math> to estimate the remainder on the subinterval <math>[-1,0]</math>. To obtain an upper bound for the remainder on <math>[0,1]</math>, we use the property <math display="inline">e^\xi <e^x</math> for <math display="inline">0<\xi<x</math> to estimate

<math display="block"> e^x = 1 + x + \frac{e^\xi}{2}x^2 < 1 + x + \frac{e^x}{2}x^2, \qquad 0 < x\leq 1 </math>

using the second order Taylor expansion. Then we solve for ''e<sup>x</sup>'' to deduce that

<math display="block"> e^x \leq \frac{1+x}{1-\frac{x^2}{2}} = 2\frac{1+x}{2-x^2} \leq 4, \qquad 0 \leq x\leq 1 </math>

simply by maximizing the [[numerator]] and minimizing the [[denominator]]. Combining these estimates for ''e<sup>x</sup>'' we see that

<math display="block"> |R_k(x)| \leq \frac{4|x|^{k+1}}{(k+1)!} \leq \frac{4}{(k+1)!}, \qquad -1\leq x \leq 1, </math>

so the required precision is certainly reached, when

<math display="block"> \frac{4}{(k+1)!} < 10^{-5} \quad \Longleftrightarrow \quad 4\cdot 10^5 < (k+1)!  \quad \Longleftrightarrow \quad k \geq 9.
</math>

(See [[factorial]] or compute by hand the values <math display="inline">9! =362880</math> and <math display="inline">10! =3628800</math>.) As a conclusion, Taylor's theorem leads to the approximation

<math display="block"> e^x = 1+x+\frac{x^2}{2!} + \cdots + \frac{x^9}{9!} + R_9(x), \qquad |R_9(x)| < 10^{-5}, \qquad -1\leq x \leq 1. </math>

For instance, this approximation provides a [[decimal representation|decimal expression]] <math>e \approx 2.71828</math>, correct up to five decimal places.

== Relationship to analyticity ==

=== Taylor expansions of real analytic functions ===

Let ''I'' ⊂ '''R''' be an [[open interval]]. By definition, a function ''f'' : ''I'' → '''R''' is [[analytic function|real analytic]] if it is locally defined by a convergent [[power series]]. This means that for every ''a''&nbsp;∈&nbsp;''I'' there exists some ''r''&nbsp;>&nbsp;0 and a sequence of coefficients ''c<sub>k</sub>''&nbsp;∈&nbsp;'''R''' such that {{nowrap|(''a'' − ''r'', ''a'' + ''r'') ⊂ ''I''}} and

<math display="block"> f(x) = \sum_{k=0}^\infty c_k(x-a)^k = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots, \qquad |x-a|<r. </math>

In general, the [[power series#Radius of convergence|radius of convergence]] of a power series can be computed from the [[Cauchy–Hadamard theorem|Cauchy–Hadamard formula]]

<math display="block"> \frac{1}{R} = \limsup_{k\to\infty}|c_k|^\frac{1}{k}. </math>

This result is based on comparison with a [[geometric series]], and the same method shows that if the power series based on ''a'' converges for some ''b'' ∈ '''R''', it must converge [[uniform convergence|uniformly]] on the [[closed interval]] <math display="inline">[a-r_b,a+r_b]</math>, where <math display="inline">r_b=\left\vert b-a \right\vert</math>. Here only the convergence of the power series is considered, and it might well be that {{nowrap|(''a'' − ''R'',''a'' + ''R'')}} extends beyond the domain ''I'' of ''f''.

The Taylor polynomials of the real analytic function ''f'' at ''a'' are simply the finite truncations

<math display="block"> P_k(x) = \sum_{j=0}^k c_j(x-a)^j, \qquad c_j = \frac{f^{(j)}(a)}{j!}</math>

of its locally defining power series, and the corresponding remainder terms are locally given by the analytic functions

<math display="block"> R_k(x) = \sum_{j=k+1}^\infty c_j(x-a)^j = (x-a)^k h_k(x), \qquad |x-a|<r. </math>

Here the functions

<math display="block">\begin{align}
& h_k:(a-r,a+r)\to \R \\[1ex]
& h_k(x) = (x-a)\sum_{j=0}^\infty c_{k+1+j} \left(x - a\right)^j
\end{align}</math>

are also analytic, since their defining power series have the same radius of convergence as the original series. Assuming that {{nowrap|[''a'' − ''r'', ''a'' + ''r'']}} ⊂ ''I'' and ''r''&nbsp;<&nbsp;''R'', all these series converge uniformly on {{nowrap|(''a'' − ''r'', ''a'' + ''r'')}}. Naturally, in the case of analytic functions one can estimate the remainder term <math display="inline">R_k(x)</math> by the tail of the sequence of the derivatives ''f′''(''a'') at the center of the expansion, but using [[complex analysis]] also another possibility arises, which is described [[Taylor's theorem#Relationship to analyticity##Taylor's theorem in complex analysis|below]].

=== Taylor's theorem and convergence of Taylor series ===

The Taylor series of ''f'' will converge in some interval in which all its derivatives are bounded and do not grow too fast as ''k'' goes to infinity. (However, even if the Taylor series converges, it might not converge to ''f'', as explained below; ''f'' is then said to be non-[[analytic function|analytic]].)

One might think of the Taylor series

<math display="block"> f(x) \approx \sum_{k=0}^\infty c_k(x-a)^k = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots </math>

of an infinitely many times differentiable function ''f'' : '''R''' → '''R''' as its "infinite order Taylor polynomial" at ''a''. Now the [[Taylor's theorem#Estimates for the remainder|estimates for the remainder]] imply that if, for any ''r'', the derivatives of ''f'' are known to be bounded over (''a''&nbsp;−&nbsp;''r'', ''a''&nbsp;+&nbsp;''r''), then for any order ''k'' and for any ''r''&nbsp;>&nbsp;0 there exists a constant {{nowrap|''M<sub>k,r</sub>'' > 0}} such that

{{NumBlk|:|<math> |R_k(x)| \leq M_{k,r} \frac{|x-a|^{k+1}}{(k+1)!} </math>|{{EquationRef|★★}}}}

for every ''x''&nbsp;∈&nbsp;(''a''&nbsp;−&nbsp;''r'',''a''&nbsp;+&nbsp;''r''). Sometimes the constants {{nowrap|''M<sub>k,r</sub>''}} can be chosen in such way that {{nowrap|''M<sub>k,r</sub>''}} is bounded above, for fixed ''r'' and all ''k''. Then the Taylor series of ''f'' [[uniform convergence|converges uniformly]] to some analytic function

<math display="block">\begin{align}
& T_f:(a-r,a+r)\to\R \\
& T_f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!} \left(x-a\right)^k
\end{align}</math>

(One also gets convergence even if {{nowrap|''M<sub>k,r</sub>''}} is not bounded above as long as it grows slowly enough.)

The limit function {{nowrap|''T<sub>f</sub>''}} is by definition always analytic, but it is not necessarily equal to the original function ''f'', even if ''f'' is infinitely differentiable. In this case, we say ''f'' is a [[non-analytic smooth function]], for example a [[flat function]]:

<math display="block">\begin{align}
& f:\R \to \R \\
& f(x) = \begin{cases}
e^{-\frac{1}{x^2}} & x>0  \\
0 & x \leq 0 .
\end{cases}
\end{align}</math>

Using the [[chain rule]] repeatedly by [[mathematical induction]], one shows that for any order&nbsp;''k'',

<math display="block"> f^{(k)}(x) = \begin{cases}
\frac{p_k(x)}{x^{3k}}\cdot e^{-\frac{1}{x^2}}  & x>0 \\
0 & x \leq 0
\end{cases}</math>

for some polynomial ''p<sub>k</sub>'' of degree 2(''k'' − 1). The function <math>e^{-\frac{1}{x^2}}</math> tends to zero faster than any polynomial as <math display="inline">x \to 0</math>, so ''f'' is infinitely many times differentiable and {{nowrap|1=''f''{{i sup|(''k'')}}(0) = 0}} for every positive integer ''k''. The above results all hold in this case:

* The Taylor series of ''f'' converges uniformly to the zero function ''T<sub>f</sub>''(''x'')&nbsp;=&nbsp;0, which is analytic with all coefficients equal to zero.
* The function ''f'' is unequal to this Taylor series, and hence non-analytic.
* For any order ''k''&nbsp;∈&nbsp;'''N''' and radius ''r''&nbsp;>&nbsp;0 there exists ''M<sub>k,r</sub>''&nbsp;>&nbsp;0 satisfying the remainder bound&nbsp;({{EquationNote|★★}}) above.
However, as ''k'' increases for fixed ''r'', the value of ''M<sub>k,r</sub>''  grows more quickly than ''r<sup>k</sup>'', and the error does not go to zero''.''

=== Taylor's theorem in complex analysis ===

Taylor's theorem generalizes to functions ''f'' : '''C''' → '''C''' which are [[complex differentiable]] in an open subset ''U''&nbsp;⊂&nbsp;'''C''' of the [[complex plane]]. However, its usefulness is dwarfed by other general theorems in [[complex analysis]]. Namely, stronger versions of related results can be deduced for [[complex differentiable]] functions ''f''&nbsp;:&nbsp;''U''&nbsp;→&nbsp;'''C''' using [[Cauchy's integral formula]] as follows.

Let ''r''&nbsp;>&nbsp;0 such that the [[closed disk]] ''B''(''z'',&nbsp;''r'')&nbsp;∪&nbsp;''S''(''z'',&nbsp;''r'') is contained in ''U''. Then Cauchy's integral formula with a positive parametrization {{nowrap|1=''γ''(''t'') = ''z'' + ''re<sup>it</sup>''}} of the circle ''S''(''z'', ''r'') with <math>t \in [0,2 \pi]</math>  gives

<math display="block">f(z) = \frac{1}{2\pi i}\int_\gamma \frac{f(w)}{w-z}\,dw, \quad f'(z) = \frac{1}{2\pi i}\int_\gamma \frac{f(w)}{(w-z)^2} \, dw, \quad \ldots, \quad f^{(k)}(z) = \frac{k!}{2\pi i}\int_\gamma \frac{f(w)}{(w-z)^{k+1}} \, dw.</math>

Here all the integrands are continuous on the [[circle]] ''S''(''z'',&nbsp;''r''), which justifies differentiation under the integral sign. In particular, if ''f'' is once [[complex differentiable]]  on the open set ''U'', then it is actually infinitely many times [[complex differentiable]] on ''U''. One also obtains [[Cauchy's estimate]]<ref>{{harvnb|Rudin|1987|loc=§10.26}}</ref>

<math display="block"> |f^{(k)}(z)| \leq \frac{k!}{2\pi}\int_\gamma \frac{M_r}{|w-z|^{k+1}} \, dw = \frac{k!M_r}{r^k}, \quad M_r = \max_{|w-c|=r}|f(w)| </math>

for any ''z''&nbsp;∈&nbsp;''U'' and ''r''&nbsp;>&nbsp;0 such that ''B''(''z'',&nbsp;''r'')&nbsp;∪&nbsp;''S''(''c'',&nbsp;''r'')&nbsp;⊂&nbsp;''U''. The estimate implies that the [[complex number|complex]] [[Taylor series]]

<math display="block"> T_f(z) =  \sum_{k=0}^\infty \frac{f^{(k)}(c)}{k!}(z-c)^k </math>

of ''f'' converges uniformly on any [[open disk]] <math display="inline">B(c,r) \subset U</math> with <math display="inline">S(c,r) \subset U</math> into some function ''T<sub>f</sub>''. Furthermore, using the [[contour integral]] formulas for the derivatives ''f''{{i sup|(''k'')}}(''c''),

<math display="block">\begin{align} 
T_f(z) &= \sum_{k=0}^\infty  \frac{(z-c)^k}{2\pi i}\int_\gamma \frac{f(w)}{(w-c)^{k+1}} \, dw \\
&=  \frac{1}{2\pi i} \int_\gamma \frac{f(w)}{w-c} \sum_{k=0}^\infty  \left(\frac{z-c}{w-c}\right)^k \, dw \\
&= \frac{1}{2\pi i} \int_\gamma \frac{f(w)}{w-c}\left( \frac{1}{1-\frac{z-c}{w-c}} \right) \, dw \\
&= \frac{1}{2\pi i} \int_\gamma \frac{f(w)}{w-z} \, dw \\
&= f(z),
\end{align}</math>

so any [[complex derivative|complex differentiable]] function ''f'' in an open set ''U''&nbsp;⊂&nbsp;'''C''' is in fact [[complex analytic]]. All that is said for real analytic functions [[Taylor's theorem#Relationship to analyticity##Taylor expansions of analytic functions|here]] holds also for complex analytic functions with the open interval ''I'' replaced by an open subset ''U''&nbsp;∈&nbsp;'''C''' and ''a''-centered intervals (''a''&nbsp;−&nbsp;''r'',&nbsp;''a''&nbsp;+&nbsp;''r'') replaced by ''c''-centered disks ''B''(''c'',&nbsp;''r''). In particular, the Taylor expansion holds in the form

<math display="block"> f(z) = P_k(z) + R_k(z), \quad P_k(z) = \sum_{j=0}^k \frac{f^{(j)}(c)}{j!}(z-c)^j, </math>

where the remainder term ''R<sub>k</sub>'' is complex analytic. Methods of complex analysis provide some powerful results regarding Taylor expansions. For example, using Cauchy's integral formula for any positively oriented [[Jordan curve]] <math display="inline">\gamma</math> which parametrizes the boundary <math display="inline">\partial W \subset U</math> of a region <math display="inline">W \subset U</math>, one obtains expressions for the derivatives {{nowrap|''f''{{i sup|(''j'')}}(''c'')}} as above, and modifying slightly the computation for {{nowrap|1=''T<sub>f</sub>''(''z'') = ''f''(''z'')}}, one arrives at the exact formula

<math display="block"> R_k(z) = \sum_{j=k+1}^\infty  \frac{(z-c)^j}{2\pi i} \int_\gamma \frac{f(w)}{(w-c)^{j+1}} \, dw = \frac{(z-c)^{k+1}}{2\pi i} \int_\gamma \frac{f(w) \, dw}{(w-c)^{k+1}(w-z)} , \qquad z\in W. </math>

The important feature here is that the quality of the approximation by a Taylor polynomial on the region <math display="inline">W \subset U</math> is dominated by the values of the function ''f'' itself on the boundary <math display="inline">\partial W \subset U</math>. Similarly, applying Cauchy's estimates to the series expression for the remainder, one obtains the uniform estimates

<math display="block"> |R_k(z)|
\leq \sum_{j=k+1}^\infty \frac{M_r |z-c|^j}{r^j}
= \frac{M_r}{r^{k+1}} \frac{|z-c|^{k+1}}{1-\frac{|z-c|}{r}}
\leq \frac{M_r \beta^{k+1}}{1-\beta}, \qquad \frac{|z-c|}{r} \leq \beta < 1. </math>

=== Example ===

[[File:Function with two poles.png|thumb|right|Complex plot of <math display="inline">f(z)=\frac{1}{1+z^2}</math>. Modulus is shown by elevation and argument by coloring: cyan&nbsp;=&nbsp;<math display="inline">0</math>, blue&nbsp;=&nbsp;<math display="inline">\frac{\pi}{3}</math>, violet&nbsp;=&nbsp;<math display="inline">\frac{2\pi}{3}</math>, red&nbsp;=&nbsp;<math>\pi</math>, yellow&nbsp;=&nbsp;<math display="inline">\frac{4\pi}{3}</math>, green&nbsp;=&nbsp;<math display="inline">\frac{5\pi}{3}</math>.]]
The function

<math display="block">\begin{align}
& f : \R \to \R \\
& f(x) = \frac{1}{1+x^2}
\end{align}</math>

is [[analytic function|real analytic]], that is, locally determined by its Taylor series. This function was plotted [[Taylor's theorem#Motivation|above]] to illustrate the fact that some elementary functions cannot be approximated by Taylor polynomials in neighborhoods of the center of expansion which are too large. This kind of behavior is easily understood in the framework of complex analysis. Namely, the function ''f'' extends into a [[meromorphic function]]

<math display="block">\begin{align}
& f:\Complex \cup \{\infty\} \to \Complex \cup \{\infty\} \\
& f(z) = \frac{1}{1+z^2}
\end{align}</math>

on the compactified complex plane. It has simple poles at <math display="inline">z=i</math> and <math display="inline">z=-i</math>, and it is analytic elsewhere. Now its Taylor series centered at ''z''<sub>0</sub> converges on any disc ''B''(''z''<sub>0</sub>, ''r'') with ''r'' < |''z''&nbsp;−&nbsp;''z''<sub>0</sub>|, where the same Taylor series converges at ''z''&nbsp;∈&nbsp;'''C'''. Therefore, Taylor series of ''f'' centered at 0 converges on ''B''(0, 1) and it does not converge for any ''z'' ∈ '''C''' with |''z''|&nbsp;>&nbsp;1 due to the poles at ''i'' and −''i''. For the same reason the Taylor series of ''f'' centered at 1 converges on <math display="inline">B(1, \sqrt{2})</math> and does not converge for any ''z''&nbsp;∈&nbsp;'''C''' with <math display="inline">\left\vert z-1 \right\vert>\sqrt{2}</math>.

== Generalizations of Taylor's theorem ==

=== Higher-order differentiability ===

A function {{math|''f'': '''R'''<sup>''n''</sup> → '''R'''}} is [[derivative|differentiable]] at {{math|'''''a''''' ∈ '''R'''<sup>''n''</sup>}} [[if and only if]] there exists a [[linear functional]] {{math|''L'' : '''R'''<sup>''n''</sup> → '''R'''}} and a function {{math|''h'' : '''R'''<sup>''n''</sup> → '''R'''}} such that

<math display="block"> f(\boldsymbol{x}) = f(\boldsymbol{a}) + L(\boldsymbol{x}-\boldsymbol{a}) + h(\boldsymbol{x})\lVert\boldsymbol{x}-\boldsymbol{a}\rVert,
\qquad \lim_{\boldsymbol{x}\to\boldsymbol{a}} h(\boldsymbol{x})=0. </math>

If this is the case, then <math display="inline">L = df(\boldsymbol{a})</math> is the (uniquely defined) [[differential of a function|differential]] of {{math|''f''}} at the point {{math|'''''a'''''}}. Furthermore, then the [[partial derivatives]] of {{math|''f''}} exist at {{math|'''''a'''''}} and the differential of {{math|''f''}} at {{math|'''''a'''''}} is given by

<math display="block"> df( \boldsymbol{a} )( \boldsymbol{v} ) = \frac{\partial f}{\partial x_1}(\boldsymbol{a}) v_1 + \cdots + \frac{\partial f}{\partial x_n}(\boldsymbol{a}) v_n. </math>

Introduce the [[multi-index notation]]

<math display="block"> |\alpha| = \alpha_1+\cdots+\alpha_n, \quad \alpha!=\alpha_1!\cdots\alpha_n!, \quad \boldsymbol{x}^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n} </math>

for {{math|''α'' ∈ '''N'''<sup>''n''</sup>}} and {{math|'''''x''''' ∈ '''R'''<sup>''n''</sup>}}.  If all the <math display="inline">k</math>-th order [[partial derivatives]] of {{math|''f'' : '''R'''<sup>''n''</sup> → '''R'''}} are continuous at {{math|'''''a''''' ∈ '''R'''<sup>''n''</sup>}}, then by [[symmetry of second derivatives|Clairaut's theorem]], one can change the order of mixed derivatives at {{math|'''''a'''''}}, so the short-hand notation

<math display="block"> D^\alpha f = \frac{\partial^{|\alpha|}f}{\partial\boldsymbol x^\alpha} = \frac{\partial^{\alpha_1 + \ldots + \alpha_n}f}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}}</math>

for the higher order [[partial derivatives]] is justified in this situation. The same is true if all the ({{math|''k'' − 1}})-th order partial derivatives of {{math|''f''}} exist in some neighborhood of {{math|'''''a'''''}} and are differentiable at {{math|'''''a'''''}}.<ref>This follows from iterated application of the theorem that if the partial derivatives of a function {{math|''f''}} exist in a neighborhood of {{math|'''''a'''''}} and are continuous at {{math|'''''a'''''}}, then the function is differentiable at {{math|'''''a'''''}}.  See, for instance, {{harvnb|Apostol|1974|loc=Theorem 12.11}}.</ref> Then we say that {{math|''f''}} is {{math|''k''}} '''times differentiable at the point&nbsp;{{math|''a''}}'''.

=== Taylor's theorem for multivariate functions ===
Using notations of the preceding section, one has the following theorem.
{{math theorem|name=Multivariate version of Taylor's theorem<ref>Königsberger Analysis 2, p. 64 ff.</ref>|math_statement= Let {{math|''f'' : '''R'''<sup>''n''</sup> → '''R'''}} be a {{math|''k''}}-times [[continuously differentiable]] function at the point {{math|'''''a''''' ∈ '''R'''<sup>''n''</sup>}}. Then there exist functions {{math|''h''<sub>''α''</sub> : '''R'''<sup>''n''</sup> → '''R'''}}, where <math>|\alpha|=k,</math> such that

<math display="block">\begin{align}
& f(\boldsymbol{x}) = \sum_{|\alpha|\leq k} \frac{D^\alpha f(\boldsymbol{a})}{\alpha!} (\boldsymbol{x}-\boldsymbol{a})^\alpha  + \sum_{|\alpha|=k} h_\alpha(\boldsymbol{x})(\boldsymbol{x}-\boldsymbol{a})^\alpha, \\
& \mbox{and}\quad \lim_{\boldsymbol{x}\to \boldsymbol{a}}h_\alpha(\boldsymbol{x})=0.
\end{align}</math>}}

If the function {{math|''f'' : '''R'''<sup>''n''</sup> → '''R'''}} is {{math|''k'' + 1}} times [[continuously differentiable]] in a [[closed ball]] <math>B = \{ \mathbf{y} \in \R^n : \left\|\mathbf{a}-\mathbf{y}\right\| \leq  r\}</math> for some <math>r > 0</math>, then one can derive an exact formula for the remainder in terms of {{nowrap|({{math|''k''+1}})-th}} order [[partial derivatives]] of ''f'' in this neighborhood.<ref>{{cite web | title = Higher-Order Derivatives and Taylor's Formula in Several Variables | last = Folland | first = G. B. | url = https://sites.math.washington.edu/~folland/Math425/taylor2.pdf | website = Department of Mathematics {{!}} University of Washington | access-date = 2024-02-21 }}</ref> Namely,

<math display="block"> \begin{align}
& f( \boldsymbol{x} ) = \sum_{|\alpha|\leq k} \frac{D^\alpha f(\boldsymbol{a})}{\alpha!} (\boldsymbol{x}-\boldsymbol{a})^\alpha  + \sum_{|\beta|=k+1} R_\beta(\boldsymbol{x})(\boldsymbol{x}-\boldsymbol{a})^\beta, \\
& R_\beta( \boldsymbol{x} ) = \frac{|\beta|}{\beta!} \int_0^1 (1-t)^{|\beta|-1}D^\beta f \big(\boldsymbol{a}+t( \boldsymbol{x}-\boldsymbol{a} )\big) \, dt.
\end{align}
</math>

In this case, due to the [[continuous function|continuity]] of ({{math|''k''+1}})-th order [[partial derivative]]s in the [[compact set]] {{math|''B''}}, one immediately obtains the uniform estimates

<math display="block"> \left|R_\beta(\boldsymbol{x})\right| \leq \frac{1}{\beta!} \max_{|\alpha|=|\beta|} \max_{\boldsymbol{y}\in B} |D^\alpha f(\boldsymbol{y})|, \qquad \boldsymbol{x}\in B. </math>

=== Example in two dimensions ===

For example, the third-order Taylor polynomial of a smooth function <math>f:\mathbb R^2\to\mathbb R</math> is, denoting <math>\boldsymbol{x}-\boldsymbol{a}=\boldsymbol{v}</math>,

<math display="block"> \begin{align}
P_3(\boldsymbol{x}) = f ( \boldsymbol{a} ) + {} &\frac{\partial f}{\partial x_1}( \boldsymbol{a} ) v_1 + \frac{\partial f}{\partial x_2}( \boldsymbol{a} ) v_2 + \frac{\partial^2 f}{\partial x_1^2}( \boldsymbol{a} ) \frac {v_1^2}{2!} +  \frac{\partial^2 f}{\partial x_1 \partial x_2}( \boldsymbol{a} ) v_1 v_2 + \frac{\partial^2 f}{\partial x_2^2}( \boldsymbol{a} ) \frac{v_2^2}{2!}  \\
& + \frac{\partial^3 f}{\partial x_1^3}( \boldsymbol{a} ) \frac{v_1^3}{3!} + \frac{\partial^3 f}{\partial x_1^2 \partial x_2}( \boldsymbol{a} ) \frac{v_1^2 v_2}{2!} + \frac{\partial^3 f}{\partial x_1 \partial x_2^2}( \boldsymbol{a} ) \frac{v_1 v_2^2}{2!} + \frac{\partial^3 f}{\partial x_2^3}( \boldsymbol{a} ) \frac{v_2^3}{3!}
\end{align}</math>

== Proofs ==

=== Proof for Taylor's theorem in one real variable ===

Let<ref>{{harvnb|Stromberg|1981}}</ref>

<math display="block"> h_k(x) = \begin{cases}
\frac{f(x) - P(x)}{(x-a)^k} & x\not=a\\
0&x=a
\end{cases}
</math>

where, as in the statement of Taylor's theorem,

<math display="block"> P(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(k)}(a)}{k!}(x-a)^k.</math>

It is sufficient to show that

<math display="block"> \lim_{x\to a} h_k(x) =0. </math>

The proof here is based on repeated application of [[L'Hôpital's rule]].  Note that, for each <math display="inline">j=0,1,...,k-1</math>, <math>f^{(j)}(a)=P^{(j)}(a)</math>.  Hence each of the first <math display="inline">k-1</math> derivatives of the numerator in <math>h_k(x)</math> vanishes at <math>x=a</math>, and the same is true of the denominator.  Also, since the condition that the function <math display="inline">f</math> be <math display="inline">k</math> times differentiable at a point requires differentiability up to order <math display="inline">k-1</math> in a neighborhood of said point (this is true, because differentiability requires a function to be defined in a whole neighborhood of a point), the numerator and its <math display="inline">k-2</math> derivatives are differentiable in a neighborhood of <math display="inline">a</math>. Clearly, the denominator also satisfies said condition, and additionally, doesn't vanish unless <math display="inline">x=a</math>, therefore all conditions necessary for L'Hôpital's rule are fulfilled, and its use is justified. So

<math display="block">\begin{align}
\lim_{x\to a} \frac{f(x) - P(x)}{(x-a)^k}
&= \lim_{x\to a} \frac{\frac{d}{dx}(f(x) - P(x))}{\frac{d}{dx}(x-a)^k} \\[1ex]
&= \cdots \\[1ex]
&= \lim_{x\to a} \frac{\frac{d^{k-1}}{dx^{k-1}}(f(x) - P(x))}{\frac{d^{k-1}}{dx^{k-1}}(x-a)^k}\\[1ex]
&= \frac{1}{k!}\lim_{x\to a} \frac{f^{(k-1)}(x) - P^{(k-1)}(x)}{x-a}\\[1ex]
&=\frac{1}{k!}(f^{(k)}(a) - P^{(k)}(a))
= 0
\end{align}</math>

where the second-to-last equality follows by the definition of the derivative at <math display="inline"> x=a</math>.

=== Alternate proof for Taylor's theorem in one real variable ===

Let <math>f(x)</math> be any real-valued continuous function to be approximated by the Taylor polynomial.

Step 1: Let <math display="inline">F</math> and <math display="inline">G</math> be functions. Set <math display="inline">F</math> and <math display="inline">G</math> to be

<math display="block">\begin{align}
F(x) = f(x) - \sum^{n-1}_{k=0} \frac{f^{(k)}(a)}{k!}(x-a)^{k} 
\end{align}</math>

<math display="block">\begin{align}
G(x) = (x-a)^{n}
\end{align}</math>

Step 2: Properties of <math display="inline">F</math> and <math display="inline">G</math>:

<math display="block">\begin{align}
F(a) & = f(a) - f(a) - f'(a)(a - a) - ... - \frac{f^{(n-1)}(a)}{(n-1)!}(a-a)^{n-1} = 0 \\
G(a) & = (a-a)^n = 0
\end{align}</math>

Similarly, 

<math display="block">\begin{align}
F'(a) = f'(a) - f'(a) - \frac{f''(a)}{(2-1)!}(a-a)^{(2-1)} - ... - \frac{f^{(n-1)}(a)}{(n-2)!}(a-a)^{n-2} = 0
\end{align}</math>

<math display="block">\begin{align}
G'(a) &= n(a-a)^{n-1} = 0\\
&\qquad \vdots\\
G^{(n-1)}(a) &= F^{(n-1)}(a) = 0
\end{align}</math>

Step 3: Use Cauchy Mean Value Theorem

Let <math>f_{1}</math> and <math>g_{1}</math> be continuous functions on <math>[a, b]</math>. Since <math>a < x < b</math> so we can work with the interval <math>[a, x]</math>. Let <math>f_{1}</math> and <math>g_{1}</math> be differentiable on <math>(a, x)</math>. Assume <math>g_{1}'(x) \neq 0</math> for all <math>x \in (a, b)</math>.
Then there exists <math>c_{1} \in (a, x)</math> such that

<math display="block">\begin{align}
\frac{f_{1}(x) - f_{1}(a)}{g_{1}(x) - g_{1}(a)} = \frac{f_{1}'(c_{1})}{g_{1}'(c_{1})}
\end{align}</math>

Note: <math>G'(x) \neq 0</math> in <math>(a, b)</math> and <math>F(a), G(a) = 0</math> so

<math display="block">\begin{align}
\frac{F(x)}{G(x)} = \frac{F(x) - F(a)}{G(x) - G(a)} = \frac{F'(c_{1})}{G'(c_{1})}
\end{align}</math>

for some <math>c_{1} \in (a, x)</math>.

This can also be performed for <math>(a, c_{1})</math>:

<math display="block">\begin{align}
\frac{F'(c_{1})}{G'(c_{1})} = \frac{F'(c_{1}) - F'(a)}{G'(c_{1}) - G'(a)} = \frac{F''(c_{2})}{G''(c_{2})}
\end{align}</math>

for some <math>c_{2} \in (a, c_{1})</math>.
This can be continued to <math>c_{n}</math>.

This gives a partition in <math>(a, b)</math>:

<math display="block">a < c_{n} < c_{n-1} < \dots < c_{1} < x</math>

with

<math display="block">\frac{F(x)}{G(x)} = \frac{F'(c_{1})}{G'(c_{1})} = \dots = \frac{F^{(n)}(c_{n})}{G^{(n)}(c_{n})} .</math>

Set <math>c = c_{n}</math>:

<math display="block">\frac{F(x)}{G(x)} = \frac{F^{(n)}(c)}{G^{(n)}(c)}</math>

Step 4: Substitute back

<math display="block">\begin{align}
\frac{F(x)}{G(x)} = \frac{f(x) - \sum^{n-1}_{k=0} \frac{f^{(k)}(a)}{k!}(x-a)^{k}}{(x-a)^{n}} = \frac{F^{(n)}(c)}{G^{(n)}(c)}
\end{align}</math>

By the Power Rule, repeated derivatives of <math>(x - a)^{n}</math>, <math>G^{(n)}(c) = n(n-1)...1</math>, so:

<math display="block">\frac{F^{(n)}(c)}{G^{(n)}(c)} = \frac{f^{(n)}(c)}{n(n-1)\cdots1} = \frac{f^{(n)}(c)}{n!}.</math>

This leads to:

<math display="block">\begin{align}
f(x) - \sum^{n-1}_{k=0} \frac{f^{(k)}(a)}{k!}(x-a)^{k} = \frac{f^{(n)}(c)}{n!}(x-a)^{n}
\end{align}.</math>

By rearranging, we get:

<math display="block">\begin{align}
f(x) = \sum^{n-1}_{k=0} \frac{f^{(k)}(a)}{k!}(x-a)^{k} + \frac{f^{(n)}(c)}{n!}(x-a)^{n}
\end{align},</math>

or because <math>c_{n} = a</math> eventually:

<math display="block">f(x) = \sum^{n}_{k=0} \frac{f^{(k)}(a)}{k!}(x-a)^{k}.</math>

=== Derivation for the mean value forms of the remainder ===

Let ''G'' be any real-valued function, continuous on the closed interval between <math display=inline>a</math> and <math display=inline>x</math> and differentiable with a non-vanishing derivative on the open interval between <math display=inline>a</math> and <math display=inline>x</math>, and define

<math display="block"> F(t) = f(t) + f'(t)(x-t) + \frac{f''(t)}{2!}(x-t)^2 + \cdots + \frac{f^{(k)}(t)}{k!}(x-t)^k.
</math>

For <math> t \in [a,x] </math>. Then, by [[mean value theorem#Cauchy's mean value theorem|Cauchy's mean value theorem]],

{{NumBlk|:|<math> \frac{F'(\xi)}{G'(\xi)} = \frac{F(x) - F(a)}{G(x) - G(a)}</math>|{{EquationRef|★★★}}}}

for some <math display="inline">\xi</math> on the open interval between <math display=inline>a</math> and <math display=inline>x</math>. Note that here the numerator <math display="inline">F(x)-F(a)=R_k(x)</math> is exactly the remainder of the Taylor polynomial for <math display="inline">y=f(x)</math>. Compute

<math display="block">\begin{align}
F'(t) = {} & f'(t) + \big(f''(t)(x-t) - f'(t)\big) + \left(\frac{f^{(3)}(t)}{2!}(x-t)^2 - \frac{f^{(2)}(t)}{1!}(x-t)\right)  +  \cdots \\
& \cdots + \left( \frac{f^{(k+1)}(t)}{k!}(x-t)^k - \frac{f^{(k)}(t)}{(k-1)!}(x-t)^{k-1}\right)
= \frac{f^{(k+1)}(t)}{k!}(x-t)^k,
\end{align}</math>

plug it into ({{EquationNote|★★★}}) and rearrange terms to find that

<math display="block"> R_k(x) = \frac{f^{(k+1)}(\xi)}{k!}(x-\xi)^k \frac{G(x)-G(a)}{G'(\xi)}.</math>

This is the form of the remainder term mentioned after the actual statement of Taylor's theorem with remainder in the mean value form.
The Lagrange form of the remainder is found by choosing <math> G(t) = (x-t)^{k+1} </math> and the Cauchy form by choosing <math> G(t) = t-a</math>.

'''Remark.''' Using this method one can also recover the integral form of the remainder by choosing

<math display="block"> G(t) = \int_a^t \frac{f^{(k+1)}(s)}{k!} (x-s)^k \, ds,</math>

but the requirements for ''f'' needed for the use of mean value theorem are too strong, if one aims to prove the claim in the case that ''f''{{i sup|(''k'')}} is only [[absolutely continuous]]. However, if one uses [[Riemann integral]] instead of [[Lebesgue integral]], the assumptions cannot be weakened.

=== Derivation for the integral form of the remainder ===

Due to the [[absolutely continuous|absolute continuity]] of <math>f^{(k)}</math> on the [[closed interval]] between <math display=inline>a</math> and <math display=inline>x</math>, its derivative <math>f^{(k+1)}</math> exists as an <math>L^1</math>-function, and we can use the [[fundamental theorem of calculus]] and [[integration by parts]]. This same proof applies for the [[Riemann integral]] assuming that <math>f^{(k)}</math> is [[continuous function|continuous]] on the closed interval and [[Differentiable function|differentiable]] on the [[open interval]] between <math display=inline>a</math> and <math display=inline>x</math>, and this leads to the same result as using the mean value theorem.

The [[fundamental theorem of calculus]] states that

<math display="block"> f(x)=f(a)+ \int_a^x \, f'(t) \, dt.</math>

Now we can [[Integration by parts|integrate by parts]] and use the fundamental theorem of calculus again to see that

<math display="block"> \begin{align}
f(x) &= f(a)+\Big(xf'(x)-af'(a)\Big)-\int_a^x tf''(t) \, dt \\
&= f(a) + x\left(f'(a) + \int_a^x f''(t) \,dt \right) -af'(a)-\int_a^x  tf''(t) \, dt \\
&= f(a)+(x-a)f'(a)+\int_a^x \, (x-t)f''(t) \, dt,
\end{align} </math>

which is exactly Taylor's theorem with remainder in the integral form in the case <math>k=1</math>. The general statement is proved using [[mathematical induction|induction]]. Suppose that
{{NumBlk|:|<math> f(x) = f(a) + \frac{f'(a)}{1!}(x - a) + \cdots + \frac{f^{(k)}(a)}{k!}(x - a)^k + \int_a^x \frac{f^{(k+1)} (t)}{k!} (x - t)^k \, dt. </math>|{{EquationRef|eq1}}}}

Integrating the remainder term by parts we arrive at

<math display="block">\begin{align}
\int_a^x \frac{f^{(k+1)} (t)}{k!} (x - t)^k \, dt = & - \left[ \frac{f^{(k+1)} (t)}{(k+1)k!} (x - t)^{k+1} \right]_a^x + \int_a^x \frac{f^{(k+2)} (t)}{(k+1)k!} (x - t)^{k+1} \, dt \\
= & \ \frac{f^{(k+1)} (a)}{(k+1)!} (x - a)^{k+1} + \int_a^x \frac{f^{(k+2)} (t)}{(k+1)!} (x - t)^{k+1} \, dt.
\end{align}</math>

Substituting this into the formula {{nowrap|in ({{EquationNote|eq1}})}} shows that if it holds for the value <math>k</math>, it must also hold for the value <math>k+1</math>. Therefore, since it holds for <math>k=1</math>, it must hold for every positive integer <math>k</math>.

=== Derivation for the remainder of multivariate Taylor polynomials ===

We prove the special case, where <math>f:\mathbb R^n\to\mathbb R</math> has continuous partial derivatives up to the order <math>k+1</math> in some closed ball <math>B</math> with center <math>\boldsymbol{a}</math>.  The strategy of the proof is to apply the one-variable case of Taylor's theorem to the restriction of <math>f</math> to the line segment adjoining <math>\boldsymbol{x}</math> and <math>\boldsymbol{a}</math>.<ref>{{harvnb|Hörmander|1976|pp=12–13}}</ref>  Parametrize the line segment between <math>\boldsymbol{a}</math> and <math>\boldsymbol{x}</math> by <math>\boldsymbol{u}(t)=\boldsymbol{a}+t(\boldsymbol{x}-\boldsymbol{a})</math> We apply the one-variable version of Taylor's theorem to the function <math>g(t) = f(\boldsymbol{u}(t))</math>:

<math display="block"> f(\boldsymbol{x})=g(1)=g(0)+\sum_{j=1}^k\frac{1}{j!}g^{(j)}(0)\ +\ \int_0^1 \frac{(1-t)^k }{k!} g^{(k+1)}(t)\, dt.</math>

Applying the [[chain rule]] for several variables gives

<math display="block">\begin{align}
g^{(j)}(t)&=\frac{d^j}{dt^j}f(\boldsymbol{u}(t))\\
&= \frac{d^j}{dt^j} f(\boldsymbol{a}+t(\boldsymbol{x}-\boldsymbol{a}))\\
&= \sum_{|\alpha| =j} \left(\begin{matrix} j\\ \alpha\end{matrix} \right) (D^\alpha f) (\boldsymbol{a}+t(\boldsymbol{x}-\boldsymbol{a})) (\boldsymbol{x}-\boldsymbol{a})^\alpha
\end{align}</math>

where <math>\tbinom j \alpha</math> is the [[multinomial coefficient]]. Since <math>\tfrac{1}{j!}\tbinom j \alpha=\tfrac{1}{\alpha!}</math>, we get:

<math display="block"> f(\boldsymbol{x})= f(\boldsymbol{a}) + \sum_{1 \leq |\alpha| \leq k}\frac{1}{\alpha!} (D^\alpha f) (\boldsymbol{a})(\boldsymbol{x}-\boldsymbol{a})^\alpha+\sum_{|\alpha|=k+1}\frac{k+1}{\alpha!}
(\boldsymbol{x}-\boldsymbol{a})^\alpha \int_0^1 (1-t)^k (D^\alpha f)(\boldsymbol{a}+t(\boldsymbol{x}-\boldsymbol{a}))\,dt.</math>

== See also ==
{{Portal|Mathematics}}
* {{annotated link|Hadamard's lemma}}
* {{annotated link|Laurent series}}
* {{annotated link|Padé approximant}}
* {{annotated link|Newton series}}
* {{annotated link|Approximation theory}}
* {{annotated link|Function approximation}}

== Footnotes ==
{{Reflist}}

== References ==
*{{citation
 | title=Calculus
 | author-link=Tom Apostol
 | first=Tom
 | last=Apostol
 | publisher=Wiley
 | year=1967
 | isbn=0-471-00005-1
 | url-access=registration
 | url=https://archive.org/details/calculus01apos
 }}.
*{{citation
 | title=Mathematical analysis
 | first=Tom
 | last=Apostol
 | publisher=Addison–Wesley
 | year=1974
}}.
*{{citation
 | first1=Robert G.
 | last1=Bartle
 | first2=Donald R.
 | last2=Sherbert
 | title=Introduction to Real Analysis
 | edition=4th
 | publisher=Wiley
 | year=2011
 | isbn=978-0-471-43331-6
}}.
*{{citation
 | first=L.
 | last=Hörmander
 | author-link=Lars Hörmander
 | title=Linear Partial Differential Operators, Volume 1
 | publisher=Springer
 | year=1976
 | isbn=978-3-540-00662-6
}}.
*{{citation
 | title = Mathematical thought from ancient to modern times, Volume 2
 | author-link=Morris Kline
 | first=Morris
 | last=Kline
 | publisher=Oxford University Press
 | year=1972
}}.
*{{citation
 | title=Calculus: An Intuitive and Physical Approach
 | first=Morris
 | last=Kline
 | publisher=Dover
 | year=1998
 | isbn=0-486-40453-6
}}.
*{{citation
 | last=Pedrick
 | first=George
 | year=1994
 | title=A First Course in Analysis
 | publisher=Springer
 | isbn=0-387-94108-8
 | url-access=registration
 | url=https://archive.org/details/firstcourseinana0000pedr
 }}.
*{{citation
 | last=Stromberg
 | first=Karl
 | title=Introduction to classical real analysis
 | publisher=Wadsworth
 | year=1981
 | isbn=978-0-534-98012-2
}}.
*{{citation
 | last=Rudin
 | first=Walter
 | title=Real and complex analysis
 | edition=3rd
 | publisher=McGraw-Hill
 | year=1987
 | isbn=0-07-054234-1
}}.
*{{citation
 | last=Tao
 | first=Terence
 | title=Analysis, Volume I
 | edition=3rd
 | publisher=Hindustan Book Agency
 | year=2014
 | isbn=978-93-80250-64-9
}}.

== External links ==
* [http://www.cut-the-knot.org/Curriculum/Calculus/TaylorSeries.shtml Taylor Series Approximation to Cosine] at [[cut-the-knot]]
* [http://cinderella.de/files/HTMLDemos/2C02_Taylor.html Trigonometric Taylor Expansion] interactive demonstrative applet
* [http://numericalmethods.eng.usf.edu/topics/taylor_series.html Taylor Series Revisited] at [http://numericalmethods.eng.usf.edu Holistic Numerical Methods Institute]

{{Calculus topics}}
{{authority control}}

[[Category:Articles containing proofs]]
[[Category:Theorems in calculus]]
[[Category:Theorems in real analysis]]
[[Category:Approximations]]