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Unformatted text preview: Power Series: 8.6/ Taylor Series 8.7 Recall from last time that a power series may converge for all x , or for no x , more generally a power series will converge in a circle. The radius of this circle is known as the radius of convergence. Theorem 6.1: A power series has three possibilities for convergence: The series converges absolutely for all real x. (i.e. The radius of conver- gence is ) The series converges ONLY for x = c (The radius of convergence is zero). The series converges absolutely for | x- c | < R and diverges for | x- c | > R. (The radius of convergence is R ). NOTE: On the radius of convergence the series must be tested. It is possible that the series converges at all points on the boundary, or that it diverge at all points on the boundary, or that it converge at some points an not at others. The obvious question to ask is whether or not a series can be differentiated....
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