Power series — Safekipedia Discoverer

In mathematics, a power series is a special kind of infinite series that looks like a long addition problem with many terms. It has the form ∑ n=0^∞ a_n (x − c)^n, where each a_n is a number called a coefficient, x is a variable, and c is a constant called the center. Power series are important because they help us understand and work with complicated infinitely differentiable functions by breaking them into simpler pieces.

When the center c is zero, the power series becomes even simpler, looking like ∑ n=0^∞ a_n x^n. These series are closely related to polynomials, which are expressions with only a few terms. In fact, the partial sums of a power series are polynomials that can approximate functions very well. Power series also appear in many other areas, such as combinatorics where they act as generating functions, and in engineering with the Z-transform.

Even everyday things like decimal notation for real numbers are examples of power series, using integer coefficients with x fixed at 1⁄10. In number theory, power series are linked to the idea of p-adic numbers. Overall, power series are a powerful tool in mathematics, helping us understand and work with many different kinds of functions and numbers.

Examples

Power series are like endless polynomials. For example, a simple polynomial like (x^2 + 2x + 3) can be written as a power series. When we center it at zero, it looks like (3 + 2x + 1x^2 + 0x^3 + 0x^4 + \dots). We can also center it at another point, like 1, and it changes shape accordingly.

The exponential function (in blue), and its improving approximation by the sum of the first n + 1 terms of its Maclaurin power series (in red). Son=0 gives f ( x ) = 1 {\displaystyle f(x)=1} ,n=1 f ( x ) = 1 + x {\displaystyle f(x)=1+x} ,n=2 f ( x ) = 1 + x + x 2 / 2 {\displaystyle f(x)=1+x+x^{2}/2} ,n=3 f ( x ) = 1 + x + x 2 / 2 + x 3 / 6 {\displaystyle f(x)=1+x+x^{2}/2+x^{3}/6} etcetera.

Another famous example is the geometric series, which looks like (\frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots). This works as long as the value of (x) stays small enough, which is called the radius of convergence. Power series help us understand many functions, like exponential growth and wave patterns, by breaking them into simpler parts.

Main article: Analytic function

Formal power series

Main article: Formal power series

In abstract algebra, formal power series help us understand the basic ideas of power series without needing real or complex numbers, or worrying about whether the series converges. This idea is very useful in a branch of math called algebraic combinatorics.

Power series in several variables

A power series can also work with more than one variable, which is useful in multivariable calculus. In this case, the series looks at how a function changes with several inputs instead of just one. The formula includes many terms, each involving products of differences between the inputs and a center point, raised to various powers.

The behavior of these series can be more complex than those with just one variable. For example, where the series settles down (converges) depends on the values of all the inputs together. There is also a way to measure the "order" of the series, which tells us the smallest combined power of the inputs that actually appears with a non-zero coefficient. This idea helps connect power series to other types of series, like Laurent series.

Main article: Multivariable calculus Main articles: Product symbol, Multi-index, Natural numbers, Tuples, Laurent series