Taylor and Maclaurin Series

A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor in 1715. If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.

It is common practice to approximate a function by using a finite number of terms of its Taylor series. Taylor's theorem gives quantitative estimates on the error in this approximation. Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial. The Taylor series of a function is the limit of that function's Taylor polynomials, provided that the limit exists. A function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an open interval (or a disc in the complex plane) is known as an analytic function.

Exponential Function as a Power Series

The exponential function (in blue) and the sum of the first $n+1$ terms of its Taylor series at $0$ (in red) up to $n=8$.

The Taylor series of a real or complex-valued function $f(x)$ that is infinitely differentiable in a neighborhood of a real or complex number $a$ is the power series:

$\begin{aligned} f(x) &= f(a)+\frac {f'(a)}{1! } (x-a)+ \frac{f''(a)}{2! } (x-a)^2+ \cdots \\ &= \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n! } \, (x-a)^{n} \end{aligned}$

where $n!$ denotes the factorial of $n$ and $f^na$ denotes the $n$th derivative of $f$ evaluated at the point $x=a$. The derivative of order zero $f$ is defined to be $f$ itself, and $(x − a)^0$ and $0!$ are both defined to be $1$. In the case that $a=0$, the series is also called a Maclaurin series.

Example 1

The Maclaurin series for $(1 − x)^{−1}$ for $\left| x \right| < 1$ is the geometric series: $1+x+x^2+x^3+\cdots\!$

so the Taylor series for $x^{−1}$ at $a=1$ is:

$1-(x-1)+(x-1)^2-(x-1)^3+\cdots$

Example 2

The Taylor series for the exponential function $e^x$ at $a=0$ is:

 $\displaystyle{1 + \frac{x^1}{1! } + \frac{x^2}{2! } + \frac{x^3}{3! } + \frac{x^4}{4! } + \frac{x^5}{5! }+ \cdots \\= 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120} + \cdots\! \\= \sum_{n=0}^\infty \frac{x^n}{n! }}$