Potenssisarjat

Potenssisarjat, or power series in English, are infinite series of the form $\sum_{n=0}^{\infty} c_n (x-a)^n$, where $c_n$ are constants called coefficients, $x$ is the variable, and $a$ is a constant known as the center of the series. These series are fundamental in calculus and analysis, playing a crucial role in approximating functions and solving differential equations.

The behavior of a power series is characterized by its radius of convergence. For a given power series, there exists a radius $R \ge 0$ such that the series converges absolutely for all $x$ satisfying $|x-a| < R$ and diverges for all $x$ satisfying $|x-a| > R$. If $R = \infty$, the series converges for all $x$. If $R = 0$, the series converges only at $x=a$. The set of $x$ values for which the series converges forms an interval called the interval of convergence.

Power series can be manipulated algebraically. They can be added, subtracted, and multiplied term by term within their interval of convergence. Differentiation and integration of power series can also be performed term by term, and the resulting series will have the same radius of convergence as the original series. This property makes power series a powerful tool for approximating complex functions with simpler polynomial forms, leading to Taylor series and Maclaurin series as special cases.