# prac3 - a ∞ X n =1 n e n b ∞ X n =1 √ n n 3-n 1 c ∞...

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MAC 3473 - PRACTICE TEST 3 Instructions: Answer all questions. Show all necessary working and reasoning. Write in such a way that any student in the class can follow your work. A Table of Integrals is supplied. Only scientiﬁc calculators are allowed. 50 points total. QUESTION 1. (2 + (4 + 4 + 4) = 14 pts) (i) Deﬁne what it means for a series X n =1 a n to converge. (ii) Determine whether the series is convergent or divergent. If it is convergent, ﬁnd its sum. ( a ) X n =1 2 n + 3 n 4 n , ( b ) X n =1 cos ± 1 n ² , ( c ) X n =1 ± cos ± 1 n ² - cos ± 1 n + 1 ²² . QUESTION 2. (4 + 4 + 4 = 12 pts) Determine whether the following series are convergent or divergent. You should state which tests you use, and your reasoning should be clear.
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Unformatted text preview: ( a ) ∞ X n =1 n ! e n , ( b ) ∞ X n =1 √ n n 3-n + 1 , ( c ) ∞ X n =2 1 (ln n ) n . QUESTION 3. (4 + 4 + 4 = 12 pts) Determine whether the following series are absolutely convergent, conditionally con-vergent or divergent. You should state which tests you use, and your reasoning should be clear. ( a ) ∞ X n =1 (-1) n arctan( n ) n 2 , ( b ) ∞ X n =2 (-1) n ln n , ( c ) ∞ X n =1 (-1) n n + 1 10 n + 1 . QUESTION 4. (2 + 10 = 12 pts) (i) What is a power series? (ii) Find the radius of convergence and the interval of convergence of the series ∞ X n =0 2 n ( x-3) n n 2 + n + 1 . 1...
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