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A power series
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{{MSC|26A09|30B10}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t0923201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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Also known as Maclaurin series. The series was published by B. Taylor in 1715, whereas a series reducible to it by a simple transformation was published by Johann I.  Bernoulli in 1694.
  
where the numerical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t0923202.png" /> is defined in some neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t0923203.png" /> and has at this point derivatives of all orders. The partial sums of a Taylor series are Taylor polynomials (cf. [[Taylor polynomial|Taylor polynomial]]).
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===One real variable===
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Let $U$ be an open set of $\mathbb R$ and consider a function $f: U \to \mathbb R$.  
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If $f$ is infinitely differentiable at $x_0$, its Taylor series at $x_0$ is the [[Power series|power series]] given by
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\begin{equation}\label{e:Taylor_series}
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\sum_{n=0}^\infty \frac{f^{(n)} (x_0)}{n!} (x-x_0)^n\, ,
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\end{equation}
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where we use the convention that $0^0=1$.  
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t0923204.png" /> is a complex number and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t0923205.png" /> is defined in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t0923206.png" /> in the field of complex numbers and is differentiable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t0923207.png" />, then there exists a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t0923208.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t0923209.png" /> is the sum of its Taylor series (1) (see [[Power series|Power series]]). If, however, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t09232010.png" /> is a real number and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t09232011.png" /> is defined in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t09232012.png" /> in the field of real numbers and has at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t09232013.png" /> derivatives of all orders, then there may be no neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t09232014.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t09232015.png" /> is the sum of its Taylor series. For instance, the function
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The partial sums
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\[
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P_k(x) := \sum_{n=0}^k \frac{f^{(n)} (x_0)}{n!} (x-x_0)^n
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\]
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of a Taylor series are called [[Taylor polynomial]] of degree $k$ and the "remainder" $f(x)- P_k (x)$ can be estimated in several ways, see [[Taylor formula]].  
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t09232016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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====Analyticity====
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The property of being infinitely differentiable does not guarantee the convergence of the Taylor series to the function $f$: a well-known example is given by the function
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\[
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f (x) =
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\left\{\begin{array}{ll}
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e^{-1/x^2} \quad &\mbox{if } x\neq 0\\
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0 \quad &\mbox{otherwise.}
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\end{array}
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\right.
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\]
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Indeed the function $f$ defined above is infinitely differentiable everywhere, its Taylor series at $0$ vanishes identically, but $f(x)>0$ for any $x\neq 0$.
  
is differentiable infinitely many times on the entire real axis, is equal to 0 only at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t09232017.png" />, but all coefficients of its Taylor series at this point are equal to 0.
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If the Taylor series of a function $f$ at $x_0$ converges to the values of $f$ in a neighborhood of $x_0$, then $f$ is [[Real analytic function|real analytic]] (in a neighborhood of $x_0$). The Taylor series is also unique in the following sense: if for some given function $f$ defined in a neighborhood of $x_0$ there is a power series $\sum a_n (x-x_0)^n$ which converges to the values of $f$, then such series coincides necessarily with the Taylor series.
  
If a function is the sum in some neighbourhood of a given point of a power series with centre at that point, then such a series is unique and is the Taylor series of this function at the given point. However, one and the same power series can be the Taylor series of different real functions. Indeed, the power series all coefficients of which are equal to 0 is the Taylor series both of the function identically equal to 0 on the entire real axis and of the function (2) at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t09232018.png" />.
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With the aid of the formulas for the difference $f (x) - P_n (x)$ (see [[Taylor formula]]) one can establish several criterions for the analyticity of $f$. A popular one is the existence of positive constants $C$, $R$ and $\delta$ such that
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\[
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|f^{(n)} (x)| \leq C n! R^n  \qquad \forall x\in ]x_0-\delta, x_0+\delta[\quad \forall n\in \mathbb N\, .
 +
\]
  
A sufficient condition for the convergence of the Taylor series (1) to the real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t09232019.png" /> on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092320/t09232020.png" /> is the existence of a common bound for all its derivatives in this interval.
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===One complex variable===
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If $U$ is a subset of the complex plane and $f:U\to \mathbb C$ an [[Holomorphic function|holomorphic function]] (i.e. complex differentiable at every point of $U$), then the Taylor series at $x_0\in U$ is given by the same formula, where $f^{(n)} (x_0)$ denotes the complex $n$-th derivative. The existence of all derivatives is guaranteed by the holomorphy of $f$, which also implies the convergence of the power series to $f$ in a neighborhood of $x_0$ (in sharp contrast with the real differentiability!), see [[Analytic function]].  
  
The Taylor series can be generalized to the case of mappings of subsets of linear normed spaces into similar spaces, and in particular to numerical functions of several variables and to functions of a matrix argument.
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===Several variables===
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The Taylor series can be generalized to functions of several variables. More precisely, if $U\subset \mathbb R^n$ and $f:U\to \mathbb R$ is infinitely differentiable at $\alpha\in U$, the Taylor series of $f$ at $\alpha$ is given by
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\begin{equation}\label{e:power_series_nd}
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\sum_{k_1, \ldots, k_n =1}^\infty \frac{1}{k_1!\ldots k_n!} \frac{\partial^{k_1+\ldots + k_n} f}{\partial x_1^{k_1} \ldots \partial x_n^{k_n}} (\alpha)\, (x_1-\alpha_1)^k_1 \ldots (x_n - \alpha_n)^{k_n}\,
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\end{equation}
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(see also [[Multi-index notation]] for other ways of expressing \eqref{e:power_series_nd}). The (real) analyticity of $f$ is defined by the property that such series converges to $f$ in a neighborhood of $\alpha$. An entirely analogous formula can be written for holomorphic functions of several variables (see [[Analytic function]]).
  
The series (1) was published by B. Taylor in 1715, a series reducible to the series (1) by a simple transformation was published by Johann I. Bernoulli in 1694.
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===Further generalizations===
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The Taylor series can be generalized to the case of mappings of subsets of linear normed spaces into similar spaces.
  
====References====
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===References===
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.A. Il'in,  V.A. Sadovnichii,  B.Kh. Sendov,  "Mathematical analysis" , Moscow  (1979)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)</TD></TR></table>
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{|
 
+
|-
 
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|valign="top"|{{Ref|Di}}|| J.A. Dieudonné,  "Foundations of modern analysis" , Acad. Press  (1960)  (Translated from French)
 
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|-
====Comments====
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|valign="top"|{{Ref|IS}}|| V.A. Il'in,  V.A. Sadovnichii,  B.Kh. Sendov,  "Mathematical analysis" , Moscow  (1979)  (In Russian)
For references see also [[Taylor formula|Taylor formula]].
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|-
 
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|valign="top"|{{Ref|Ni}}|| S.M. Nikol'skii,  "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)
====References====
+
|-
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.A. Dieudonné,  "Foundations of modern analysis" , Acad. Press  (1960)  (Translated from French)</TD></TR></table>
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|valign="top"|{{Ref|Ru}}|| W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78 {{MR|0385023}} {{ZBL|0346.26002}}
 +
|-
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|valign="top"|{{Ref|St}}|| K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)
 +
|-
 +
|}

Revision as of 09:39, 27 December 2013

2020 Mathematics Subject Classification: Primary: 26A09 Secondary: 30B10 [MSN][ZBL]

Also known as Maclaurin series. The series was published by B. Taylor in 1715, whereas a series reducible to it by a simple transformation was published by Johann I. Bernoulli in 1694.

One real variable

Let $U$ be an open set of $\mathbb R$ and consider a function $f: U \to \mathbb R$. If $f$ is infinitely differentiable at $x_0$, its Taylor series at $x_0$ is the power series given by \begin{equation}\label{e:Taylor_series} \sum_{n=0}^\infty \frac{f^{(n)} (x_0)}{n!} (x-x_0)^n\, , \end{equation} where we use the convention that $0^0=1$.

The partial sums \[ P_k(x) := \sum_{n=0}^k \frac{f^{(n)} (x_0)}{n!} (x-x_0)^n \] of a Taylor series are called Taylor polynomial of degree $k$ and the "remainder" $f(x)- P_k (x)$ can be estimated in several ways, see Taylor formula.

Analyticity

The property of being infinitely differentiable does not guarantee the convergence of the Taylor series to the function $f$: a well-known example is given by the function \[ f (x) = \left\{\begin{array}{ll} e^{-1/x^2} \quad &\mbox{if } x\neq 0\\ 0 \quad &\mbox{otherwise.} \end{array} \right. \] Indeed the function $f$ defined above is infinitely differentiable everywhere, its Taylor series at $0$ vanishes identically, but $f(x)>0$ for any $x\neq 0$.

If the Taylor series of a function $f$ at $x_0$ converges to the values of $f$ in a neighborhood of $x_0$, then $f$ is real analytic (in a neighborhood of $x_0$). The Taylor series is also unique in the following sense: if for some given function $f$ defined in a neighborhood of $x_0$ there is a power series $\sum a_n (x-x_0)^n$ which converges to the values of $f$, then such series coincides necessarily with the Taylor series.

With the aid of the formulas for the difference $f (x) - P_n (x)$ (see Taylor formula) one can establish several criterions for the analyticity of $f$. A popular one is the existence of positive constants $C$, $R$ and $\delta$ such that \[ |f^{(n)} (x)| \leq C n! R^n \qquad \forall x\in ]x_0-\delta, x_0+\delta[\quad \forall n\in \mathbb N\, . \]

One complex variable

If $U$ is a subset of the complex plane and $f:U\to \mathbb C$ an holomorphic function (i.e. complex differentiable at every point of $U$), then the Taylor series at $x_0\in U$ is given by the same formula, where $f^{(n)} (x_0)$ denotes the complex $n$-th derivative. The existence of all derivatives is guaranteed by the holomorphy of $f$, which also implies the convergence of the power series to $f$ in a neighborhood of $x_0$ (in sharp contrast with the real differentiability!), see Analytic function.

Several variables

The Taylor series can be generalized to functions of several variables. More precisely, if $U\subset \mathbb R^n$ and $f:U\to \mathbb R$ is infinitely differentiable at $\alpha\in U$, the Taylor series of $f$ at $\alpha$ is given by \begin{equation}\label{e:power_series_nd} \sum_{k_1, \ldots, k_n =1}^\infty \frac{1}{k_1!\ldots k_n!} \frac{\partial^{k_1+\ldots + k_n} f}{\partial x_1^{k_1} \ldots \partial x_n^{k_n}} (\alpha)\, (x_1-\alpha_1)^k_1 \ldots (x_n - \alpha_n)^{k_n}\, \end{equation} (see also Multi-index notation for other ways of expressing \eqref{e:power_series_nd}). The (real) analyticity of $f$ is defined by the property that such series converges to $f$ in a neighborhood of $\alpha$. An entirely analogous formula can be written for holomorphic functions of several variables (see Analytic function).

Further generalizations

The Taylor series can be generalized to the case of mappings of subsets of linear normed spaces into similar spaces.

References

[Di] J.A. Dieudonné, "Foundations of modern analysis" , Acad. Press (1960) (Translated from French)
[IS] V.A. Il'in, V.A. Sadovnichii, B.Kh. Sendov, "Mathematical analysis" , Moscow (1979) (In Russian)
[Ni] S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)
[Ru] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78 MR0385023 Zbl 0346.26002
[St] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)
How to Cite This Entry:
Taylor series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Taylor_series&oldid=15684
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article