Interval and Radius of Convergence for Power Series

Find the interval and radius of convergence for the power series representation of $\displaystyle \frac{1}{1+x^2}$.

Finding the interval and radius of convergence for a power series is a fundamental topic in calculus, particularly within the study of infinite series. Power series are expressions of functions as infinite sums, and understanding their convergence properties is essential in both theoretical and applied mathematics. The interval of convergence is the set of all points where the series converges, and the radius of convergence is the distance from the center of the series to the boundary points of this interval. These concepts are crucial when working with series because they dictate the range of input values for which the series representation of a function is valid.

To solve this particular problem, you would typically start by identifying the power series associated with the given function. In this case, the function is 1 divided by the sum of 1 and the square of x, a classic geometric series when expressed in terms of a variable squared. You would apply methods such as the ratio test or root test to determine where the series converges absolutely. These tests provide a systematic way to assess the behavior of the series' terms as they extend towards infinity.

Understanding these principles can greatly enhance your ability to work with series and representations of functions, which have significant applications in mathematical modeling, physics, and engineering. Power series offer a versatile approach to approximating functions and solving differential equations, which is why mastering convergence concepts is a critical learning objective in advanced mathematics courses.