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Home Infinite Series Power series Corollary: Interval of Convergence  
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Corollary: Interval of Convergence
COROLLARY For each power series , one of the following happens. (i) The series converges absolutely at x = 0 and diverges everywhere else. (ii) The series converges absolutely on the whole real line (∞, ∞). (iii) The series converges absolutely at every point in an open interval (r, r) and diverges at every point outside the closed interval [r, r]. At the endpoints r and r the series may converge or diverge, so the interval of convergence is one of the sets (r,r), [r, r), (r, r], [r, r]. Figure 9.7.1 illustrates part (iii) of the Corollary. The number r is called the radius of convergence of the power series. In case (i) the radius of convergence is zero, and in case (ii) it is ∞. Once the radius of convergence is determined, we need only test the series at x = r and x = r to find the interval of convergence. Figure 9.7.1
If we replace x by x  c we obtain a power series in x  c, The power series has the same radius of convergence as and the interval of convergence is simply moved over so that its center is c instead of 0. For example, if has interval of convergence ( r, r], then has interval of convergence (c  r, c + r], illustrated in Figure 9.7.2. Figure 9.7.2


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