Quiz 2 Problem 1. [10 points] Let S consist of all positive integers with no prime factor larger than 3, and define X Thus, the first few terms of the sum are X 1-6 12 (a)Write a closed-form expression in the box that makes the equation below true. X j=0k=0 Solution. Every positive integer with no prime factor larger than 3 has the form 3 for some nonnegative integers j and k. Thus, the expression 23k makes the equation true (b) Write a closed-form expression in the box that makes this equation true X Solution. Well apply the formula for an infinite geometric sum twice 27 j=0k=0 1-1/3 1-1 3�� Quiz 2 3 Problem 1. [10 points] Let S consist of all positive integers with no prime factor larger than 3, and define: � 1 X = k k∈S Thus, the first few terms of the sum are: 1 1 1 1 1 1 1 1 X = + + + + + + + + . . . 1 2 3 4 6 8 9 12 (a) Write a closed­form expression in the box that makes the equation below true. ∞ ∞ X = j=0 k=0 Solution. Every positive integer with no prime factor larger than 3 has the form 2j3k for some nonnegative integers j and k. Thus, the expression 1 2j3k makes the equation true. (b) Write a closed­form expression in the box that makes this equation true: X = Solution. We’ll apply the formula for an infinite geometric sum twice. � � �∞ �∞ 1 �∞ 1 �∞ 1 = 2j3k 2j 3k j=0 k=0 j=0 �∞ 1 k=0 � 1 � = j=0 � 2j 1 1 − 1/3 ��∞ 1 = 1 − 1/3 j=0 � � � 2j � 1 1 = 1 − 1/3 1 − 1/2 = 3