Math 132 Exam 3 Fall 2016 15 multiple choice questions worth 5 points each 2 hand graded questions worth 12 and 13 points each Exam covers sections 11 1 11 6 Sequences Series Integral Comparison Alternating Absolute not Root or Ratio No calculators For the multiple choice questions mark your answer on the answer card Show all your work for the written problems Your ability to make your solution clear will be part of the grade Useful Formulas Pn i 1 Pn i 1 i3 Pn n n 1 2 i n n 1 2 i 1 2 i2 n n 1 2n 1 6 sin2 cos2 1 1 tan2 sec2 1 cot2 csc2 sin A B sin A cos B sin B cos A cos A B cos A cos B sin A sin B tan A B cos A cos B R 1 2 tan A tan B 1 tan A tan B cos A B cos A B sin A sin B 1 2 cos A B cos A B sin A cos B 1 2 sin A B cos A B sin2 x 12 1 cos 2x cos2 x 21 1 cos 2x sin 2 2 sin cos cos 2 cos2 sin2 csc x dx ln csc x cot x C R sec x dx ln sec x tan x C Math 132 Exam 3 Page 2 of 20 1 Determine whether the sequence defined by an ln 2n3 2 ln 5n3 2n2 4 converges or diverges If it converges find the limit A 0 B ln 2 5 C ln D 2 5 2 5 E 2 F 5 G Diverges Solution lim an lim ln 2n3 2 ln 5n3 2n2 4 n n 2n3 2 lim ln n 5n3 2n2 4 2n3 2 ln lim n 5n3 2n2 4 2 ln 5 2 Find ALL possible values of x for which the series X 9 xn n 0 5n converges A It is not possible to find such x because the series diverges B x 0 C x 1 D 3 x 3 E 3 x 5 F 5 x 5 Math 132 Exam 3 Page 3 of 20 G x Solution X 9 xn n 0 5n X X xn 9 n 5 5n n 0 n 0 n X X x n 1 9 5 5 n 0 n 0 These are both geometric series The first has r 1 5 so it converges The second has r x 5 so it converges when x5 1 or 5 x 5 Math 132 Exam 3 Page 4 of 20 3 Determine whether the sequence defined by 2 2 an n cos n2 2 converges or diverges If it converges find the limit A 2 B 1 C 0 D 1 E 2 F G Diverges Solution 2 lim an lim n cos n n n2 2 cos n22 2 Apply L Hopital s Rule lim n 1 n2 sin n22 2 4 n3 lim n 2 n3 2 lim 2 sin 2 sin 2 2 n n2 2 2 4 Which of the following sequences converge an 2n 1 n 4 bn A dn only B an bn only C cn dn only D an dn only n n100 cn ln n10 n dn n4 n 1 Math 132 Exam 3 Page 5 of 20 E an bn dn only F an cn dn only G All of them Solution For an write out the factorial and see what cancels 2n 1 2n 2n 1 n 5 n 4 n 3 1 2n 1 n 4 n 4 n 3 n 2 1 2n 1 2n 2n 1 n 5 Diverges an For bn n n100 you can apply L Hopital s Rule 100 times to get ln n n n 100 lim bn lim n For cn apply L Hopital ln n10 10 ln n lim Apply L Hopital n n n n1 2 n 10 n 20 lim lim 1 2 0 1 2 n 1 2 n n n lim cn lim For dn consider what happens when n 8 n4 n4 n 1 n 1 n n 1 n 2 n 3 1 n4 n 2 n 2 n 2 n 2 n 3 1 4 n4 n 1 1 0 0 n 2 4 n 3 n 2 n 3 dn Math 132 Exam 3 5 Assume the terms of a sequence an are given by the following formula an 1 22 32 n2 3n3 3n3 3n3 3n3 Find the limit of the sequence or conclude that it diverges A 0 B 1 D 1 2 2 3 E 1 6 C 1 9 G Diverges F Solution We use the formula on the formula sheet 22 32 n2 1 3 lim an lim n n 3n3 3n3 3n3 3n 1 lim 1 2 2 3 2 n2 3 n 3n n X n n 1 2n 1 1 1 2 lim i lim n 3n3 n 3n3 6 i 1 1 n n 1 2n 1 n 18n3 9 lim 6 Determine the value of the series X n 2 6 n n 3 or conclude that it diverges A 0 B 13 2 Page 6 of 20 Math 132 Exam 3 C D E F Page 7 of 20 5 3 13 6 13 3 10 2 G Diverges Solution This is telescoping so you need to use partial fractions to write out the series 6 2 2 n n 3 n n 3 X n 2 X 2 2 6 n n 3 n 2 n n 3 2 2 2 2 2 2 2 2 2 2 2 5 3 6 4 7 5 8 6 9 So SN 1 2 2 2 2 2 3 4 N 2 N 3 N 4 and lim SN 1 n 2 2 13 3 4 6 Math 132 Exam 3 Page 8 of 20 7 Determine the value of the series X 2n 2 3n 1 4n n 0 or conclude that it diverges A 4 C 9 2 13 2 D 37 4 B E F 25 2 97 8 G Diverges Solution This is two geometric series X X X 2n 2 3n 1 2n 2 3n 1 4n 4n 4n n 0 n 0 n 0 n n X X 1 1 3 3 4 2 4 n 0 n 0 1 4 3 1 1 2 1 3 4 25 1 12 2 2 8 Assume X n 1 A B C D E 22 5 2 5 1 10 3 10 92 an is an infinite series with partial sums given by SN 4 2 N What is a5 Math 132 Exam 3 F 52 1 G 10 3 H 10 Solution Note that a5 S5 S4 2 4 5 2 2 1 1 4 4 5 2 10 Page 9 of 20 Math 132 Exam 3 Page 10 of 20 9 Which of the following series converge I X 1 n n 1 1 II 1 n X n 2 1 ln n4 III X n 1 n n2 1 …
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