Power series and radius of convergence from "summary" of Differential Calculus by S Balachandra Rao
A power series is an infinite series of the form Σan(x - c)n, where an represents the coefficients, x is the variable, and c is a constant. These series are a fundamental tool in mathematics, allowing us to represent functions as an infinite sum of powers of the variable x. The key to understanding power series lies in the concept of convergence. The radius of convergence of a power series is the distance from the center point c at which the series converges. It is defined as the largest real number R such that the series converges for all x satisfying |x - c| < R and diverges for |x - c| > R. The radius of convergence can be determined using the ratio test, which involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If this limit is less than 1, the series converges; if it is greater than 1, the series diverges; and if it is equal to 1, the test is inconclusive. It is important to note that the behavior of a power series at the endpoints of its interval of convergence can vary. At points where |x - c| = R, the series may converge or diverge, depending on the specific function being represented. Therefore, it is crucial to test the endpoints separately to determine the overall convergence of the series. In summary, power series provide a powerful tool for representing functions as infinite sums of powers of the variable x. The radius of convergence determines the interval over which the series converges, and careful analysis is needed to understand its behavior at the endpoints. By studying power series and their radius of convergence, we can gain valuable insights into the behavior of functions and solve complex mathematical problems.
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