hatványsor

A hatványsor, or power series in English, is a fundamental concept in mathematics, particularly in calculus and analysis. It is an infinite series of the form $\sum_{n=0}^{\infty} a_n (x-c)^n$, where $a_n$ are constants representing the coefficients of the series, and $c$ is a constant representing the center of the series. The variable $x$ is the independent variable.

The behavior of a power series is often studied in terms of its convergence. A power series converges for certain values of $x$. The set of all values of $x$ for which a power series converges forms an interval, known as the interval of convergence. This interval is centered at $c$ and has a radius, called the radius of convergence, denoted by $R$. If $R$ is finite and non-zero, the series converges for $|x-c| < R$ and diverges for $|x-c| > R$. Special cases include $R=0$ (converges only at $x=c$) and $R=\infty$ (converges for all $x$).

Power series are incredibly useful for approximating functions. Many important functions, such as exponential, trigonometric, and logarithmic functions, can be represented by their Maclaurin series (a power series centered at $c=0$) or Taylor series (a power series centered at an arbitrary $c$). These series provide a way to express complex functions as infinite sums of simple polynomial terms, which are easier to manipulate and analyze. They are essential tools in fields like differential equations, physics, and engineering.