Techniques for the Finding Interval of Convergence See the table below for the four combinations. The series could converge or diverge at each of the two endpoints. Here, there are a number of possibilities as well. Any x-values greater than a distance of R from c would cause the series to diverge.īut what about those points that are exactly R units away? That is, c – R and c + R? In other words, there would be a finite positive number R, called the radius of convergence, such that the series converges for all x-values that are within R units from the center c. When a series has a finite interval of convergence, then it’s always centered at… well… the center! Now, that second point is probably the trickiest. The converges only at the center x = c, diverging at every other x-value.And the series diverges outside of that interval. There may be a limited range of x-values, called the interval of convergence, for which the series converges. There are three things that could happen. In fact, the series may converge (have a finite sum) for some values of x, but diverge at others.įor more about convergence and divergence in general, check out: AP Calculus BC Review: Series Convergence. Now anytime you have an infinite series (infinitely many terms), you have to worry about issues of convergence. Recall that a power series, with center c, is a series of functions of the following form. And we’ll also see a few examples similar to those you might find on the AP Calculus BC exam. This article reviews the definitions and techniques for finding radius and interval of convergence of power series. Whenever you work with a power series, you have to be careful about its radius and interval of convergence.
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