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Final exam solutions

# 3 this series converges 1 n n log n 1 n log n and 1 n

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3. This series converges: 1 n + n log n 1 n log n , and 1 n log n converges by the integral test. 4

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Problem 5. 30 points. Suppose that f ( x ) is a diﬀerentiable function on R , suppose that f (0) = 0, f (1) = 1, f (2) = 1. Prove that there exists an x (0 , 2) such that f 0 ( x ) = 1 / 5. State which theorems you are using. Proof. Let g ( x ) = f ( x ) - x/ 5. Then g (0) = 0, g (1) = 4 / 5 and g (2) = 3 / 5. The function g ( x ) is continuous, hence by the Intermediate Value Theorem there is a a (0 , 1) such that g ( a ) = 3 / 5. By Rolle’s theorem there is an x ( a, 2) such that g 0 ( x ) = 0. Then f 0 ( x ) = 1 / 5. 5
Problem 6. 30 points. Consider the sequence of functions f n ( x ) = x - x n deﬁned on [0 , 1]. 1. Show that f n ( x ) converges pointwise to a function f ( x ) on [0 , 1], and ﬁnd a formula for f ( x ). 2. Does f n ( x ) converge to f ( x ) uniformly on [0 , 1]? Justify. 3. Does f n ( x ) converge to f ( x ) uniformly on [0 ,a ] for a < 1? Justify. Solution

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3 This series converges 1 n n log n 1 n log n and 1 n log n...

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