The radius of convergence of a power series is the distance from the center of the series to the nearest point where the series converges. It is a measure of how far the series can be extended before it diverges.
To find the radius of convergence, we use the ratio test, which compares the absolute value of the terms in the series to a geometric series. If the limit of this ratio is less than 1, then the series converges and the radius of convergence is the value of x for which this limit is equal to 1. If the limit is greater than 1, then the series diverges.
The interval of convergence is the range of values for x in which the power series converges. It is determined by the radius of convergence, with the interval being centered at the center of the series and extending to the left and right by the radius value.
No, the radius of convergence is always a positive value. It represents a distance and therefore cannot be negative.
At the endpoints of the interval, the series may converge or diverge. To determine this, we can use the endpoints test, which involves plugging in the value of x at the endpoint into the original series. If the series converges at the endpoint, then it is included in the interval of convergence. If the series diverges at the endpoint, then it is not included in the interval of convergence.