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# What is the radius of convergence of a power series? What is the interval of convergence of a power series?

Solution

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A power series is an infinite series

\begin{aligned} a_0+a_1(x-c)+a_2(x-c)^2+\cdots=\sum_{n=0}^{\infty} a_n(x-c)^n. \end{aligned}

The point $c$ is called expansion point.

• A power series converges absolutely in a symmetric interval about its expansion point, and diverges outside that symmetric interval. The distance from the expansion point to an endpoint is called the radius of convergence.

• Any combination of convergence or divergence may occur at the endpoints of the interval. That is, the series may diverge at both endpoints, converge at both endpoints, or diverge at one and converge at the other.

• A power series always converges at the expansion point. The set of points where the series converges is called the interval of convergence.

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