Interval and Radius of Convergence

A power series is a mathematical representation of a function as an infinite sum of terms involving powers of a variable. Such series converge only for specific input values, making it essential to determine the range over which the series produces valid results. This leads to the concepts of radius and interval of convergence, which define where the series behaves meaningfully.

The radius of convergence describes the distance from the center within which the power series converges. For a general power series centered at a, this can be written as:

\begin{equation*}\sum\limits_{n=0}^{\infty} c_n (x - a)^n\end{equation*}

A standard method to determine this radius is the ratio test. The limit of the ratio of consecutive terms is:

\begin{equation*}L = \lim_{n \to \infty} \left| \jfrac{c_{n+1}(x - a)^{n+1}}{c_n (x - a)^n} \right|\end{equation*}

This simplifies to:

\begin{equation*}L = |x - a| \lim_{n \to \infty} \left| \jfrac{c_{n+1}}{c_n} \right|\end{equation*}

For convergence, the condition L < 1 must hold, which gives:

\begin{equation*}|x - a| \lim_{n \to \infty} \left| \jfrac{c_{n+1}}{c_n} \right|<1\end{equation*}

Rearranging this inequality yields the radius of convergence:

\begin{equation*}R = \lim_{n \to \infty} \left| \frac{c_n}{c_{n+1}} \right|\end{equation*}

The interval of convergence is the set of all real values of x that satisfy:

\begin{equation*}|x-a|<R\end{equation*}

This defines the open interval (a - R, a + R). However, the endpoints x = a ± R must be tested separately, since the ratio test is inconclusive at these boundary points.