Taylor series

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[edit]

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[edit]

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[edit]

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[edit]

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 =0}^\infty \frac{1}{k_1!\cdots k_n!} \frac{\partial^{k_1+\cdots + k_n} f}{\partial x_1^{k_1} \cdots \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[edit]

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

References[edit]