Explanation Theorem used: Suppose the power series is given by ∑ n = 0 ∞ a n ( x − a ) n and let R = lim n → ∞ | a n a n + 1 | . a. If R = 0 , then the series converges only for x = a . b. If 0 < R < ∞ , the series converges for x in the interval ( a − R , a + R ) and diverges for x outside the interval [ a − R , a + R ] . c. If R = ∞ , the series converges for all x. Calculation: The given power series is ∑ n = 0 ∞ ( 2 n ) ! x n . Here, a n = ( 2 n ) ! . By using the above mentioned theorem, the radius of convergence of the power series ∑ n = 0 ∞ ( 2 n ) ! x n is R = lim n → ∞ | ( 2 n ) ! ( 2 n + 2 ) ! | . Solve R = lim n → ∞ | ( 2 n ) ! ( 2 n + 2 ) ! | as follows

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Author: Tan

Publisher: INTER CENG

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Chapter 11.5, Problem 8E

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To find: The radius of convergence and the interval of convergence of the power series ∑n=0∞(2n)!xn.

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Chapter 11.5, Problem 7E

Chapter 11.5, Problem 9E

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Chapter 11 Solutions

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Ch. 11.1 - In Exercises 110, find the first three Taylor...Ch. 11.1 - Prob. 10E Ch. 11.1 - Prob. 11E Ch. 11.1 - Prob. 12E Ch. 11.1 - Prob. 13E Ch. 11.1 - Prob. 14E Ch. 11.1 - In Exercises 1120, find the nth Taylor polynomial...Ch. 11.1 - Prob. 16E Ch. 11.1 - Prob. 17E Ch. 11.1 - Prob. 18E Ch. 11.1 - Prob. 19E Ch. 11.1 - Prob. 20E Ch. 11.1 - Prob. 21E Ch. 11.1 - Prob. 22E Ch. 11.1 - Prob. 23E Ch. 11.1 - Prob. 24E Ch. 11.1 - Prob. 25E Ch. 11.1 - Prob. 26E Ch. 11.1 - Prob. 27E Ch. 11.1 - Prob. 28E Ch. 11.1 - Prob. 29E Ch. 11.1 - Prob. 30E Ch. 11.1 - Prob. 31E Ch. 11.1 - Prob. 32E Ch. 11.1 - Prob. 33E Ch. 11.1 - Prob. 34E Ch. 11.1 - Let f(x) = ex. a. 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The radius of convergence and the interval of convergence of the power series ∑ n = 0 ∞ ( 2 n ) ! x n .

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To find: The radius of convergence and the interval of convergence of the power series ∑n=0∞(2n)!xn.

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Chapter 11 Solutions