A power series is an infinite series of the form ∑(a_n * (x – c)^(n)), where a_n are coefficients, x is the variable, c is a constant, and the summation is taken over all non-negative integers n. Power series play a crucial role in calculus and mathematical analysis, as they can be used to represent a wide variety of functions, including exponential, logarithmic, trigonometric, and other transcendental functions.
Power series can be manipulated and analyzed using techniques from calculus, such as differentiation and integration. They are often used in conjunction with other methods, such as the Taylor series and the Maclaurin series, to approximate functions and solve differential equations. The radius of convergence of a power series determines the interval within which the series converges to the function it represents.
Power series are also essential in the study of complex analysis, where they are used to define analytic functions and investigate their properties. Moreover, power series are used in many applications, such as physics, engineering, and finance, to model and analyze various phenomena.