Taylor Series · aiwiki.page

A Taylor series is an infinite series constructed from the values of a function and its successive derivatives at a fixed point. It is a particular power series, with coefficients determined by differentiation. When the series converges to the original function, it represents that function through an infinite sum of polynomial terms. Its finite partial sums provide local approximations, but the existence of derivatives of every order does not by itself guarantee an exact series representation. (openstax.org)

Definition and coefficients

Suppose $f$ has derivatives of every order in a neighborhood of $a$ . Its Taylor series centered at $a$ is

\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n = f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots.

Here $f^{(0)}=f$ , and $n!$ denotes the factorial, with $0!=1$ . The center $a$ is also called the expansion point. When $a=0$ , the expansion is called a Maclaurin series. (openstax.org)

The Taylor polynomial of order $N$ is

T_N(x)=\sum_{n=0}^{N}\frac{f^{(n)}(a)}{n!}(x-a)^n.

It has degree at most $N$ and matches $f$ and its first $N$ derivatives at $a$ . This matching uniquely determines the polynomial: differentiating a power series $n$ times and evaluating at its center isolates $n!$ times its $n$ th coefficient. Consequently, any convergent power-series representation of $f$ around $a$ must be its Taylor series. (openstax.org)

Taylor’s theorem and approximation error

Taylor’s theorem relates finite Taylor polynomials to the function without requiring an infinite expansion. If $f$ has continuous derivatives through order $N+1$ on an interval containing $a$ and $x$ , then

f(x)=T_N(x)+R_N(x),

where the Lagrange form of the remainder is

R_N(x)= \frac{f^{(N+1)}(\xi)}{(N+1)!}(x-a)^{N+1}

for some $\xi$ between $a$ and $x$ . For $N=0$ , this is the mean value theorem. (openstax.org)

If $|f^{(N+1)}(t)|\le M$ throughout that interval, then

|R_N(x)|\le \frac{M|x-a|^{N+1}}{(N+1)!}.

This estimate quantifies the error introduced by truncation. Equality between the infinite Taylor series and $f(x)$ requires the limit $R_N(x)\to0$ as $N\to\infty$ . Accuracy for a fixed finite order near the center and convergence as the order increases are therefore distinct questions. (openstax.org)

Convergence and analyticity

Like every power series, a Taylor series has a radius of convergence $R$ , possibly zero or infinite. It converges absolutely for $|x-a|<R$ and diverges for $|x-a|>R$ . For real variables, behavior at the endpoints must be checked separately. Changing the center can change both the coefficients and the interval of convergence. (openstax.org)

An analytic function agrees locally with its Taylor series. Analyticity is stronger than infinite differentiability. A standard counterexample is

f(x)= \begin{cases} e^{-1/x^2},&x\ne0,\\ 0,&x=0. \end{cases}

This function has derivatives of every order, all zero at the origin. Its Maclaurin series is therefore identically zero, although $f(x)>0$ whenever $x\ne0$ . The series converges everywhere but represents the function only at zero. (math.ucdavis.edu)

In complex analysis, a function holomorphic on a disk has a Taylor representation throughout that disk. The nearest obstruction to holomorphic continuation determines the maximal convergence radius. Thus complex singularities can limit a real-variable expansion even when the function is smooth on the entire real axis. For example, the expansion of $1/(1+x^2)$ at zero has radius $1$ , reflecting singularities at $i$ and $-i$ . (dlmf.nist.gov)

Standard expansions

The exponential function and the sine and cosine functions have Maclaurin expansions

e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}, \qquad \sin x=\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!},

\cos x=\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}.

All three converge for every real or complex argument. Trigonometric arguments use radians. (openstax.org)

The geometric series gives

\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n, \qquad |x|<1.

Integrating the corresponding expansion for $1/(1+x)$ yields

\log(1+x)= \sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^n}{n}.

For real $x$ , this last series represents the logarithm on $-1<x\le1$ ; it diverges at $x=-1$ . These examples distinguish expansions with unlimited convergence from those valid only on restricted domains. (openstax.org)

Operations and applications

Within the convergence interval, power series can be differentiated and integrated term by term. These operations preserve the radius of convergence, although endpoint convergence can change. This makes Taylor expansions useful in calculus for evaluating an integral or constructing series solutions of a differential equation. Finite truncations also support function approximation in numerical analysis. (lemesurierb.people.charleston.edu)

For functions of several variables, Taylor expansions use partial derivatives. The second-order local model has the form

f(\mathbf a+\mathbf h)\approx f(\mathbf a)+\nabla f(\mathbf a)^{T}\mathbf h +\frac12\mathbf h^{T}H_f(\mathbf a)\mathbf h,

where $\nabla f$ is the gradient and $H_f$ the Hessian matrix. Linear and quadratic models underpin Newton’s method and related methods in mathematical optimization. (ocw.mit.edu)

Historical development

The series is named after Brook Taylor, whose Methodus incrementorum directa et inversa appeared in 1715. Related expansions had already been developed by mathematicians including James Gregory, Isaac Newton, and Gottfried Wilhelm Leibniz. Taylor’s work presented a general formulation within the developing calculus, rather than introducing all the underlying ideas for the first time. (mathshistory.st-andrews.ac.uk)