Explanation Given series is x ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 + ∑ n = 0 ∞ a n x n . Simplify the series as follows. x ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 + ∑ n = 0 ∞ a n x n = ∑ n = 2 ∞ n ( n − 1 ) x a n x n − 2 + ∑ n = 0 ∞ a n x n = ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 + 1 + ∑ n = 0 ∞ a n x n = ∑ n = 2 ∞ n ( n − 1 ) a n x n − 1 + ∑ n = 0 ∞ a n x n Since the power of x is n – 1 in the series ∑ n = 2 ∞ n ( n − 1 ) a n x n − 1 , consider a variable k = n − 1 . Then the value of n is k + 1. Now substitute the value of n for the first series of x ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 + ∑ n = 0 ∞ a n x n as follows. x ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 + ∑ n = 0 ∞ a n x n = ∑ k + 1 = 2 ∞ ( k + 1 ) ( k + 1 − 1 ) a k + 1 x k − 1 − 1 + ∑ n = 0 ∞ a n x n = ∑ k = 2 − 1 ∞ ( k + 1 ) k a

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ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 5.1, Problem 27P

To determine

The series x∑n=2∞n(n−1)anxn−2+∑n=0∞anxn as a sum with the generic term involving xn.

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Chapter 5.1, Problem 26P

Chapter 5.1, Problem 28P

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The series x ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 + ∑ n = 0 ∞ a n x n as a sum with the generic term involving x n .

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The series x∑n=2∞n(n−1)anxn−2+∑n=0∞anxn as a sum with the generic term involving xn.

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