A power series is an infinite series of the form $\sum_{n=0}^{\infty} a_n(x-a)^n$, where $a_n$ are constants and $x$ is the variable. It is a type of mathematical series that is commonly used in calculus and other branches of mathematics.
A Taylor series is a type of power series that is specifically centered around a particular value of $x$ (usually denoted by $a$). This means that the variable $x$ is replaced by $(x-a)$ in the power series formula. In other words, a Taylor series is a special case of a power series.
Power series are used in calculus to represent functions as infinite sums of simpler polynomial terms. By manipulating and evaluating these series, we can approximate the values of functions, find their derivatives and integrals, and solve differential equations.
The convergence of a power series refers to whether or not the series will approach a finite limit as the number of terms approaches infinity. This depends on the values of the coefficients $a_n$ and the variable $x$. If the series converges, it can be used to approximate the value of the function for a given $x$ value.
To determine the convergence of a power series, we can use various convergence tests such as the ratio test, root test, and integral test. These tests help us determine the range of $x$ values for which the series will converge. In some cases, the convergence may also depend on the values of the coefficients $a_n$.