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Unformatted text preview: is one of the intervals ( aR,a + R ) , ( aR,a + R ] , [ aR,a + R ) , or [ aR,a + R ] . Remark 1.2. Knowing the radius of convergence is a positive real number, R , does not tell you whether the series converges or diverges when  xa  = R . Example 1.3 (Example of a Bessel Function. These equations ±rst arose in solving Kepler’s equation to describe planetary motion) . Find the radius and interval of convergence. J ( x ) = ∞ ± n =0 (1) n x 2 n 2 2 n ( n !) 2 Section 11.8 Power Series 3 Example 1.4. (problem 6) Find the radius and interval of convergence. ∞ ± n =1 √ nx n Example 1.5. (problem 18) Find the radius and interval of convergence. ∞ ± n =1 n 4 n ( x + 1) n Example 1.6. (problem 20) Find the radius and interval of convergence. ∞ ± n =1 (3 x2) n n · 3 n...
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 Fall '07
 Zhang
 Calculus, Power Series, Radius of convergence, Cauchy–Hadamard theorem

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