Nov 11, 2025 · In this section we will give the definition of the power series as well as the definition of the radius of convergence and interval of convergence for a power series.

In this section we define power series and show how to determine when a power series converges and when it diverges. We also show how to represent certain functions using power series.

Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. All holomorphic functions are complex-analytic.

Most power series, however, converge for more than one value of x. In that case, the power series either converges for all real numbers x or converges for all x in a finite interval.

A geometric series is a special case of a power series with a constant coefficient. Convergence happens when the common ratio's absolute value is less than 1. The interval of convergence is the range of x …

We summarize the results and convergence tests that we use to study power series. Power series are closely related to geometric series, so most of the tests involve comparisons with a geometric series.

Nov 1, 2025 · To determine the interval or radius of convergence for a power series, you must apply one of the convergence tests learned in previous concepts, and find the conditions on x that make …

Apr 13, 2022 · Since on the interior of the disk of convergence the ratio and root tests tell us that the series converges absolutely, you might wonder if on the edges the series either diverges or …

Find the radius and interval of convergence for each series. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence.

We typically use the ratio test or the root test to determine the convergence of power series. With these tests, we know a series will converge if our computed limit results in ρ <1.