Explanation Given: Chinese people suggested replacing one-child policy by one-son policy. In both policies, natural birth rate is same. Result used: The sum of the geometric series is ∑ k = 0 ∞ a r k = a 1 − r if | r | < 1 . Calculation: The one-child policy states, each couple have one child. Compute the number of children under the one-son policy. Suppose the first child is son, then the probability is 1 2 . Suppose the first child is daughter and the second child is son, then the probability is ( 1 2 ) 2 . In general, the first n − 1 children are girls and the n th a boy, the probability is ( 1 2 ) n . Thus, the total number of children under the one-son policy is ∑ i = 1 ∞ i ⋅ ( 1 2 ) i . Evaluate the series ∑ i = 1 ∞ i ⋅ ( 1 2 ) i . Consider f ( x ) = ∑ i = 1 ∞ i ⋅ x i (1) If f ( x ) = ∑ i = 1 ∞ i ⋅ x i , then f ( x ) + ∑ i = 1 ∞ x i = ∑ i = 1 ∞ i x i + ∑ i = 1 ∞ x i . That is, f ( x ) + ∑ i = 1 ∞ x i = ∑ i = 1 ∞ ( i + 1 ) x i . Let g ( x ) = ∑ i = 1 ∞ x i + 1 . The function g ( x ) can be written as g ( x ) = − 1 − x + ∑ i = 0 ∞ x i . Thus, by the Result stated above, g ′ ( x ) = − 1 − x + 1 1 − x [ where r = x and a = 1 ]

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Chapter 8.3, Problem 83E

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To compare: The total number of children under the two policies using geometric series.

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Chapter 8.3, Problem 82E

Chapter 8.3, Problem 84E

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To compare: The total number of children under the two policies using geometric series.

Solution Summary: The author compares the total number of children under the two policies using geometric series. The one-child policy states, each couple have one child.

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