n_13490 - x = a ± R must be checked with some other test 1...

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§ 11.8 Power Series 1.Theorem3 (p.742) For a given power series n =0 c n ( x - a ) n there are only three possibilities: (i) The series converges only when x = a ( R = 0). (ii) The series converges for all x ( R = ). (iii) There is a positive number R such that the series converges if | x - a | < R and diverges if | x - a | > R . Remark: (i) In general, the Ratio Test and Root Test should be use to determine the radius of convergence R and R = 1 L . (ii) The Ratio Test and Root Test always fail when x is an endpoint of the interval of convergence, so the endpoints
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Unformatted text preview: x = a ± R must be checked with some other test. 1 Important Limits 1. lim n →∞ (1 + 1 n ) n = e 2. lim n →∞ n √ n = 1 2 11.8.16 Find the radius of convergence and interval of convergence of the series. ∞ X n =1 (-1) n x 2 n-1 (2 n-1)! 3 11.8.20 Find the radius of convergence and interval of convergence of the series. ∞ X n =1 (3 x-2) n n 3 n 4...
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