The definition of convergence for a power series is the limit of the partial sums of the series as the number of terms approaches infinity. If this limit exists, then the series is said to converge.
The radius of convergence can be determined by using the ratio test or the root test. These tests compare the limit of the absolute value of the ratio of consecutive terms to a certain value, and if the limit is less than one, the series will converge. The radius of convergence is then equal to the value at which the limit is equal to one.
Yes, a power series can converge at its endpoints. This occurs when the series converges at the value of the variable where the series is centered. For example, if a series is centered at x=2, it can converge at both x=2 and x=4 (since 4 is 2 units away from the center).
The radius of convergence is the distance from the center of the power series to the nearest point where the series converges. The interval of convergence is the range of values for the variable where the series converges. The radius of convergence determines the endpoints of the interval of convergence.
Yes, a power series can diverge at some points within its interval of convergence. This is because the radius of convergence only determines the endpoints of the interval, and the series may converge or diverge at other points within the interval. This can be determined by using other tests such as the alternating series test or the integral test.