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Tut5_March2 - a ∞ µ n =1 4(27 n 2 3 e ∞ µ n =1 3 n 2...

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MATH 1005A Tutorial 5 March 2, 2007 Instructor: Dr. A. Alaca 1. Find the limit of the given sequence if it converges. Otherwise, state why it diverges. a) a n =1+( - 1) n b) a n = n 2 e n c) a n = n sin(1 /n ) d) a n = ( 1+ 2 n ) 1 /n 2. Determine whether each of the following sequences is monotonic, has an upper bound, a lower bound and a limit? a) ± 3 , ² 3 3 , ³ 3 ² 3 3 , ··· ´ . (Ans: The limit is 3). b) a 1 =7 ,a n +1 =15+ a n - 2 ,n 1 . (Ans: The limit is (31 + 53) / 2.) 3. Suppose that µ a n = 5 and s n is the n th partial sum of the series. What is lim n -→∞ a n ? What is lim n -→∞ s n ? 4. Show that if µ n =1 a n converges, then µ n =1 (1 /a n ) diverges. 5. If µ n =2 a 1+ a · n = 3, and a> 0, determine the value of a . Ans: a = 3+ 21 2 . 6. Find the sum of the series: a) µ n =1 3 n +2 7 n b) µ n =1 ( - 5) n +1 2 3 n c) µ n =1 1 n 2 +5 n +6 7. Use the integral test to determine whether the series is convergent or divergent. a) µ n =1 n e n 2 b) µ n =2 1 n ln n c) µ n =2 1 n (ln n ) 2 d) µ n =1 n ( n 2 +1) 2 8. Determine whether or not the given series converges. Justify your answer.
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Unformatted text preview: a) ∞ µ n =1 4 (27 n ) 2 / 3 e) ∞ µ n =1 3 n 2 + 1 2 n 4-1 b) ∞ µ n =1 n 3 2 n 3 + 1 f) ∞ µ n =1 sin n n 2 c) ∞ µ n =1 4 n 2 + n 3 √ n 7 + n 3 g) ∞ µ n =1 (-1) n n 3 n 3 + 7 d) ∞ µ n =2 1 ln n h) ∞ µ n =2 1 (ln n ) 2 9. Which of the following series converges? (I) ∞ µ n =1 n n + 1 (II) ∞ µ n =1 π n 3 n (III) ∞ µ n =1 1 n √ n (a) I and III (b) II and III (c) II (d) III 10. Which of the following series converges? (I) ∞ µ n =1 1 e n (II) ∞ µ n =1 1 √ e n (III) ∞ µ n =1 1 3 √ e n (a) I and II (b) II and III (c) I and III (d) I, II and III 11. What is the sum of the series ∞ µ n =1 3 n +1 4 n ? a) 6 b) 8 c) 9 d) 12....
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