The interval of convergence is the range of values for which a given series will converge, meaning that the sum of its terms will approach a finite value as the number of terms increases.
To determine the interval of convergence, you can use tests such as the ratio test, root test, or alternating series test. These tests help determine the radius of convergence, which is the distance between the center of the series and the point where the series converges. The interval of convergence will then be the range of values within this radius.
The radius of convergence is the distance between the center of the series and the point where the series converges. It is a crucial value in determining the interval of convergence, as it helps identify the range of values for which the series will converge.
Determining the interval of convergence is essential because it helps us understand the behavior of a given series. It tells us the range of values for which the series will converge and whether the series will diverge for values outside of this interval.
If a series does not have a finite interval of convergence, it means that the series will either diverge for all values or converge for all values. In such cases, it is necessary to use more advanced techniques, such as power series or Laurent series, to determine the behavior of the series.