M413test3s - Test 3 Solutions 1 State the definition of a...

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Unformatted text preview: Test 3 Solutions 1 . State the definition of a metric d on a set S . See Ross, definition 13.1. 2 . Find the interior and boundary of the set { ( x, y ) ∈ R 2 : x 2 + y 2 ≤ 1 } . The interior is { ( x, y ) ∈ R 2 : x 2 + y 2 < 1 } . The boundary is { ( x, y ) ∈ R 2 : x 2 + y 2 = 1 } 3 . For the infinite series ∞ n =1 2 n 2 + 2 n (a) Find the n-th partial sum s n . Hint: 2 n 2 + 2 n = 1 n- 1 n + 2 . (b) Find the sum of the series. We have s n = n k =1 2 k 2 + 2 k = n k =1 1 k- 1 k + 2 = 1 1- 1 3 + 1 2- 1 4 + 1 3- 1 5 + 1 4- 1 6 + ··· + 1 n- 2- 1 n + 1 n- 1- 1 n + 1 + 1 n- 1 n + 2 = 1 + 1 2- 1 n + 1- 1 n + 2 . So the sum of the series is s = lim n →∞ s n = lim n →∞ 1 + 1 2- 1 n + 1- 1 n + 2 = 1 + 1 2 = 3 2 . 4 . Use the integral test to determine if ∞ n =1 n n 2 + 1 converges. We must determine if ∞ 1 x x 2 + 1 dx converges. Now ∞ 1 x x 2 + 1 dx = lim t →∞ t 1 x x 2 + 1 dx ....
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This note was uploaded on 08/24/2009 for the course MATH 262447221 taught by Professor Weisbart during the Spring '09 term at UCLA.

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M413test3s - Test 3 Solutions 1 State the definition of a...

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