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. Description: The function series is ∑ n = 0 ∞ a n ( x − a ) n . The given statement can be disproved by the following counter example. Suppose that ∑ n = 0 ∞ a n ( x − a ) n has radius of convergence R. Let ∑ n = 0 ∞ 1 2 n ( x − 1 ) n . Here, a n = 1 2 n . By using the above mentioned theorem, the radius of convergence of the power series ∑ n = 0 ∞ 1 2 n ( x − 1 ) n is R = lim n → ∞ | 1 2 n 1 2 n + 1 | . Solve R = lim n → ∞ | 1 2 n 1 2 n + 1 | as follows. R = lim n → ∞ | 1 2 n 1 2 n + 1 | = lim n → ∞ ( 1 2 n × 2 n + 1 1 ) = lim n → ∞ ( 1 2 n × 2 n ⋅ 2 1 ) = 2 Thus, the radius of convergence of the power series ∑ n = 0 ∞ 1 2 n ( x − 1 ) n is R = 2

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Chapter 11.5, Problem 36E

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Whether the statement “If ∑n=0∞an(x−a)n has radius of convergence R, then ∑n=0∞an2(x−a)n has radius of convergence R” is true or false.

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Chapter 11.5, Problem 35E

Chapter 11.6, Problem 1CQ

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Whether the statement “If ∑ n = 0 ∞ a n ( x − a ) n has radius of convergence R , then ∑ n = 0 ∞ a n 2 ( x − a ) n has radius of convergence R ” is true or false.

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Whether the statement “If ∑n=0∞an(x−a)n has radius of convergence R, then ∑n=0∞an2(x−a)n has radius of convergence R” is true or false.

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Chapter 11 Solutions