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

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Chapter 5.1, Problem 21P

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

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Chapter 5.1, Problem 20P

Chapter 5.1, Problem 22P

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

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

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