Power series

A power series is a mathematical tool used to represent functions as an infinite sum of terms. Each term in the series consists of a coefficient, which is a constant number, multiplied by a variable raised to a specific power. The series also features a fixed starting point known as the center, denoted as $c$. You can think of a power series as a generalized polynomial. While a standard polynomial has a finite number of terms, a power series continues forever.

To understand how they work, imagine a function that describes a smooth, continuous curve. Mathematicians often want to approximate these complex curves using simpler shapes. By using a Taylor series, which is a specific type of power series, we can create polynomial approximations of a function near its center. As we add more terms from the series, the polynomial gets closer to the actual function. If a function can be represented by a convergent power series in a neighborhood around a point, we call that function analytic.

Power series follow strict rules regarding the exponents they use. In a standard power series, the exponents must be non-negative integers. This means you cannot have negative powers, such as $x^{-1}$, which would instead form a Laurent series. You also cannot have fractional powers, like $x^{1/2}$, which are found in Puiseux series. Additionally, the coefficients must be constants that do not depend on the variable $x$. These constraints ensure the series maintains its specific structure.

Not every power series will provide a meaningful value for every possible number. A series might only work for values of $x$ that are close to the center $c$. The distance from the center to the edge of this working zone is called the radius of convergence, often denoted as $R$. If the distance from the center is less than $R$, the series converges to a specific value. If the distance is greater than $R$, the series diverges, meaning it does not settle on a single number. The collection of all points that allow the series to converge forms a shape called the disc of convergence.

Determining this radius is a key part of mathematical analysis. The Cauchy–Hadamard theorem provides a way to calculate this radius using the coefficients of the series. Specifically, it relates the radius to the limit superior of the $n$-th root of the absolute value of the coefficients. While the series is well-behaved inside the disc of convergence, the behavior at the boundary can be unpredictable. At the boundary, a series might converge at some points and diverge at others, or even result in a discontinuous sum.

Mathematicians can perform various operations on power series to create new ones. If you have two power series, you can add or subtract them by combining their corresponding terms. The radius of convergence for the new sum will be at least as large as the smaller of the two original radii. You can also multiply two series using a method called the Cauchy product. For division, mathematicians use recursive methods to solve for the terms of the resulting series. Furthermore, power series can be differentiated or integrated term by term.

Beyond basic calculus, power series appear in many specialized fields. In combinatorics, they are used as generating functions to study sequences and numerical patterns. This is particularly helpful in analytic combinatorics for estimating the size of various structures. Electronic engineers utilize a version of these series known as the Z-transform. Even our decimal number system is a form of power series, where the coefficients are integers and the variable is fixed at ten.

Finally, the theory extends into even more complex territory with multivariable power series. These series use multiple variables at once, which is essential for multivariable calculus. Instead of a simple interval, the region of convergence for these series can be a more complex shape. In abstract algebra, mathematicians study formal power series. These allow researchers to explore the essence of power series without worrying about whether the series actually converges.