dcrisci said:
The radius of convergence is a measure of how closely a power series approximates a function. It is the distance from the center of the series to the nearest point where the series converges.
The radius of convergence is determined by applying the ratio test to the power series. This involves taking the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th and nth terms of the series. If this limit is less than 1, the series converges and the radius of convergence is the reciprocal of this limit. If the limit is greater than 1, the series diverges and the radius of convergence is 0. If the limit is exactly 1, further tests are needed to determine the convergence of the series.
The interval of convergence is the set of all values of x for which the power series converges. It is centered around the point x = a, which is the center of the series.
The interval of convergence is determined by testing the endpoints of the interval for convergence or divergence. If the power series converges at the endpoints, those values are included in the interval of convergence. If the series diverges at the endpoints, those values are not included in the interval of convergence. The interval may also be determined by considering the radius of convergence and the behavior of the function at the endpoints.
The radius and interval of convergence help determine the accuracy and range of a power series as an approximation for a function. They are also important for determining the domain of convergence for a power series, which can have practical applications in fields such as engineering and physics.
