M138.assn6.W07 - MATH 138 Assignment 6(2 pages Winter 2007...

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Unformatted text preview: MATH 138 Assignment 6 (2 pages) Winter 2007 Submit all problems marked * on Friday, Nlarch 2 Note: 1 . This assignment consists of problems on sequences, taken from Chapter 4 of your Course Notes. and problems on the introductory lectures on series, from your text and elsewhere. Due to Test 1, this assignment is longer than usual. Try to do some of the problems on your study break; otherwise it will be heavy going when you return! . The arc length problem #8 b) from Assignment 5. a) Problem 1, page 58 *b) Problem 3, page 58 (1) Problem 6, page 62 (1) Problem 8, page 62 *0) Problem 12, page 63 In problems 2 e), d), e), and 3, you are looking for the ‘cut oil" value of n needed to prove the given inequality. . a) Problem 13, page 66 *b) Problem 17, page 66 (3) Problem 20, page 66 a) Problem 23, page 68 *b) Problem 24, page 68 . These problems involve the use of the Limit Theorems 1 and 2 on pages 68 and 70. a) Problem 28, page 7] b) Problem 31, page 71 These problems involve the use of induction and the l\'lonotone Sequence Theorem. a) Problem 3, page 77 b) Problem 5, page 77 *c) Problem 7. page 77 [Suggestion in 0), try a conjecture of the form 0 < J:,,+1 < :1:.,,, < 2.] Use the 5 — 6 definition of lim f(:1:) = L to prove each limit. Ill—'0' l *a) lim (3sz — 1) = — 1)) lim 1:2 = 0 0) lim fl = [l w—vQ 2 r—vO .r—»0 . Use the diagram on page 65 of your Course Notes to show that i sin 0: — sin ,6] g |ov — Bl. llence use the E - 6 definition of limit to prove that lim since 2 sin a, i.e., than f(a;') = sin 1: is a m—>a continuous function at each a 6 IR. Find the sum of the given geometric series. 00 00 TL * 0° 1 17- 0° 7r" 3+2" * 6 EL) ;3(_1) b) Z2371~1 C) E 3n+2 d) Zgn—l 11:0 71:0 1 1.: 1 . A super ball, falling from a height y onto a hard flat surface, rebounds to height ry , where 0 < r < 1. if the ball is initially dropped from a height H and continues to bounce indefinitely. find the total distance it travels. MATH 138 - Winter 2007 Assignment #6 Page 2 of 2 11. For each of the following, either prove the statement is TRUE, or give a. coui'iter—exzunple which shows that it is FALSE. a) If 11111 run 2: 00 and lim yn = 3), then lim ."L'n'yn = 00. Til—*OO Tl—’OO H.400 *b) If neither {33”} nor {yn} Converges, then {mg/n} does not converge. *c) If {|9:,,|} converges, then {xn} converges. u: d) If the set {an} has a. limit as n —> 00, then the series E a", converges. 11:1 DC 0) If the series E a.” diverges. then sequence {an} has no limit as w, —> oo. 11:] CHALLENGES: Here are some problems about interesting sets: the Cantor Set, the Sierpinski Carpet and the Snowflake Curve. 12. Problem 65 on page 722 of your text. 13. Problem 5 on page 789 of your text. ...
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