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Convergence Theorem for Power Series If series diverges for converges for , then it converges absolutely for all , then it diverges for all . If the DEFINITION: RADIUS AND INTERVAL OF CONVERGENCE Possible behaviour of . 1. There is an (radius of convergence) such that the series diverges for but converges absolutely for . The series may or may not converge at the endpoints and . Interval of Convergence : or 2. The series converges absolutely for every . ( ) 3. The series converges at and diverges elsewhere. Example. Determine the radius and interval of convergence of the power series Solution: Using Ratio test, we get For the series to converge, we should have as suggested by Ratio test. So So, the radius of convergence . The interval of convergence is the solution set of w/c is included because they make the series a divergent p-series with . . End points are not