Since absolute convergence implies convergence the

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Unformatted text preview: e function ∞ an xn . f (x) = a0 + a1 x + a2 x2 + a3 x3 + . . . + an xn + . . . = n=0 But n an xn converges absolutely on (−R, R) and diverges for |x| > R when 0 < R < ∞. Since absolute convergence implies convergence, the series thus converges everywhere on (−R, R). To check what happens at the end-points |x| = R of the interval (−R, R) we’ll need to use those other convergence tests. When 0 < R < ∞, therefore, the interval of convergence of a power series n an xn is one of the four intervals (−R, R) , [−R, R) , (−R, R] , [−R, R] . And when R = 0 the interval of convergence is just the point x = 0, while if R = ∞ the interval of convergence is all of (−∞, ∞). So the Radius of convergence and all those convergence tests tell us where functions f (x) = n an xn are defined! Example 2: find the interval of convergence of the power series ∞ n=0 Now at x = 1 4 the series becomes ∞ (−4)n n √ x....
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