Final exam solutions

# Prove that lim s n t n proof we need to show that for

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real number. Prove that lim( s n + t n ) = + . Proof. We need to show that for every real number M there exists a natural number N such that s n + t n > M for all n > N . Since inf( t n ) = a , we know that t n a for all n . Fix M . We know from lim s n = + that there exists a natural number N such that s n > M - a for all n > N . But then s n + t n > M for all n > N . 3

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Problem 4. 30 points. For each of these series, state whether or not it converges. Explain which convergence test you are using. 1. n 2 3 n 2. n =2 ( - 1) n sin n 3. 1 n + n log n . 1. This series converges by the ratio test: lim a n +1 a n = lim ( n + 1) 2 3 n 2 = 1 3 . 2. This series diverges, because ( - 1) n n 1 . 3. This series converges: 1 n + n log n 1 n log n , and 1 n log n converges by the integral test. 4
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.

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