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 ser...
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