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Unformatted text preview: MATH 74 HOMEWORK 5 (DUE WEDNESDAY OCTOBER 3) 1. Prove that for any n N one has n j =1 j 2 j = ( n- 1)2 n +1 + 2. 2. The n th derivative f ( n ) ( x ) of a function f ( x ) can be defined recursively by f (1) ( x ) = f ( x ) and f ( n +1) ( x ) = d dx f ( n ) ( x ) , n N . Fix f ( x ) = xe x . Find, and prove by induction, an explicit formula for f ( n ) ( x ) for any n N . [Assume all of calculus in this exercise.] Exercises 3-6. You will work with inequalities in these exercises. Assume it is known that for any real numbers a, b, c, p we have IE1 if a b , then a + c b + c . IE2 if a b and c 0, then ac bc , IE3 if a b and c < 0, then ac bc . IE4 if 0 < a b , then 1 b 1 a . IE5 if 0 a < b and p > 0, then a p < b p . Whenever you deduce a new inequality from an old one, it is likely you are using one of these facts. You do not need to cite these facts explicitly in your work....
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- Fall '07