Exam3_212_Fall2023_FreeResponce

The City College of New York The Department of Mathematics Math 21200– Exam #3 Instructor: Samuel Magill 13 NOVEMBER 2023 Name : EmpID : Directions. Please read each question carefully, show all work even for the multiple choice, and check afterwards that you have answered all of each question correctly. Important: No books, calculators, cell phones, computers, laptops or notes are al- lowed. You must show all your work to receive credit. Any crossed out work will be disregarded (even if correct). Write one clear answer with a coherent derivation for each question. You have 100 minutes to complete this exam. The multiple choice counts as 70 percent of the exam 2 grade and the free response counts for 30 percent of the exam 2 grade. The Free response is used to reassess the continuity portion of the exam 1 grade. 1

Math 21200 Exam #3 Name: 1. (20 points) Find an explicit formula for the sequence of partial sums to determine if the series converges. If the series converges, determine what the series converges to (a) ª§¦ª êçæêôæ n = 1 1 n ( n + 1 ) (b) ª§¦ª êçæêôæ n = 2 ( √ n + 1 − √ n ) 2. (10 points) State the monotone convergence theorem. Define all important words used. Page 2 of 14

Math 21200 Exam #3 Name: 4. (15 points) Find the interval of convergence of the power series ª§¦ª êçæêôæ n = 0 n ( x + 2 ) n 5 n √ n 2 + 4 Remember to check the endpoints if applicable. 5. (15 points) Define g ( x ) via the power series g ( x ) = ª§¦ª êçæêôæ n = 0 x n ( n + 1 ) 3 . Compute g ′ ( − 1 2 ) (be sure to defend why your computation is valid) and approx- imate g ′ ( − 1 2 ) with an error less than 0.01. Hint: For the second part the alternating estimation theorem is of help. Page 4 of 14

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 1) 2 1 · 2 + 2 2 · 3 + 2 3 · 4 + ... + 2 n(n + 1) + ... 1) A) 2(n + 2) n + 1 ; 2 B) 2(n + 1) n + 2 ; 2 C) 2(n + 1) n ; 2 D) 2n n + 1 ; 2 Find the sum of the series. 2) _ Q n = 0 ( - 1) n 4 9 n 2) A) 1 2 B) 2 5 C) 18 5 D) 9 2 Use the nth - Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. 3) _ Q n = 1 ln 3 n 3) A) diverges B) converges; ln 3 C) converges; ln 1 3 D) inconclusive 4) _ Q n = 1 n n + 8 4) A) converges, 8 B) inconclusive C) diverges D) converges, 1 8 Determine if the series converges or diverges. If the series converges, find its sum. 5) _ Q n = 1 1 ln(n + 4) - 1 ln(n + 5) 5) A) converges; ln 5 B) diverges C) converges; 1 ln 4 D) converges; 1 ln 5 1

10) Which of the following statements is false? 10) A) The integral test does not apply to divergent sequences. B) a _ a If a n and f(n) satisfy the requirements of the Integral Test, and if Q 1 f(x)dx converges, then Q n = 1 a n = Q 1 f(x) dx. C) _ Q n = 2 1 n(ln n) p converges if p > 1. D) _ Q n = 1 1 n p converges if p > 1 and diverges if p K 1. 11) Which of the following sequences do not meet the conditions of the Integral Test? I. a n = n(sin n + 1) II. a n = 1 n p + p III. a n = 1 n n 11) A) I, II, and III B) II and III C) I and III D) I only 12) Which of the following statements is false? 12) A) _ _ If {a n } and {b n } meet the conditions of the Limit Comparison test, then, if lim n ±Q a n b n = 0 and b n converges, then a n converges. B) All of these are true. C) The sequences {a n } and {b n } must be positive for all n to apply the Limit Comparison Test. D) _ The series a n must have no negative terms in order for the Direct Comparison test to be applicable. 13) _ Suppose that a n > 0 and b n > 0 for all n L N (N an integer). If lim n ±Q a n b n = Q , what can you conclude about the convergence of a n ? 13) A) _ _ a n converges if b n converges B) _ _ _ _ a n diverges if b n diverges, and a n converges if b n converges C) The convergence of a n cannot be determined. D) _ _ a n diverges if b n diverges 14) _ If the series Q n = 0 ( - 1) n (x - 2) n is integrated twice term by term, for what value(s) of x does the new series converge? Be mindful of endpoints if applicable. 14) A) < x < B) < x K C) K x K D) K x < 3

15) _ Find the sum of the series Q n = 1 ( - 1) n - 1 n 3 n - 1 by expressing 1 1 + x as a geometric series, differentiating both sides of the resulting equation with respect to x, and replacing x by 1 3 . 15) A) 16 9 B) 9 16 C) 4 9 D) 9 4 Find the Maclaurin series for the given function. 16) e - 2x 16) A) _ Q n = 1 2 n x n n! B) _ Q n = 0 2 n x n n! C) _ Q n = 1 ( - 1) n 2 n x n n! D) _ Q n = 0 ( - 1) n 2 n x n n! 17) 1 5 + x 17) A) _ Q n = 1 ( - 1) n x n 5 n + 1 B) _ Q n = 0 ( - 1) n x n + 1 5 n C) _ Q n = 0 ( - 1) n x n 5 n + 1 D) _ Q n = 0 ( - 1) n x n 5 n 18) sin 9x x 18) A) _ Q n = 1 ( - 1) 2n + 1 9 2n + 1 x 2n (2n + 1)! B) _ Q n = 1 ( - 1) n 9 2n + 1 x 2n (2n + 1)! C) _ Q n = 0 ( - 1) n 9 2n + 1 x 2n (2n + 1)! D) _ Q n = 0 ( - 1) 2n + 1 9 2n + 1 x 2n (2n + 1)! 19) cos ( - 6x) 19) A) _ Q n = 0 ( - 1) n 6 2n x 2n (2n)! B) _ Q n = 0 ( - 1) 2n 6 2n x 2n (2n)! C) _ - Q n = 0 ( - 1) 2n 6 2n x 2n (2n)! D) _ - Q n = 0 ( - 1) n 6 2n x 2n (2n)! 4

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine if the series converges or diverges. If the series converges, find its sum. 1) _ Q n = 1 1 ln(n + 4) - 1 ln(n + 5) 1) A) diverges B) converges; 1 ln 4 C) converges; 1 ln 5 D) converges; ln 5 Use the nth - Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. 2) _ Q n = 1 ln 5 n 2) A) inconclusive B) converges; ln 5 C) converges; ln 1 5 D) diverges 3) _ Q n = 1 n n + 7 3) A) diverges B) inconclusive C) converges, 1 7 D) converges, 7 Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 4) 7 1 · 2 + 7 2 · 3 + 7 3 · 4 + ... + 7 n(n + 1) + ... 4) A) 7(n + 1) n + 2 ; 7 B) 7(n + 1) n ; 7 C) 7n n + 1 ; 7 D) 7(n + 2) n + 1 ; 7 Find the sum of the series. 5) _ Q n = 0 ( - 1) n 5 7 n 5) A) 5 6 B) 5 8 C) 35 6 D) 35 8 1

Determine if the series converges or diverges; if the series converges, find its sum. 6) _ Q n = 0 cos n Δ 2 n 6) A) Converges; 2 3 B) Converges; 2 C) Converges; 1 D) Diverges Find the values of x for which the geometric series converges. 7) _ Q n = 0 ( - 1) n x - 8 9 n 7) A) - 10 < x < 26 B) - 17 < x < 17 C) - 1 < x < 17 D) 10 < x < 26 8) _ Q n = 0 sin 6x n 8) A) diverges for all x B) x x is not an odd multiple of Δ /12 C) - Q < x < Q D) x x is not a multiple of Δ Provide an appropriate response. 9) Which of the following is not a condition for applying the integral test to the sequence {a n }, where a n = f(n)? I. f(x) is everywhere positive II. f(x) is decreasing for x L N III. f(x) is continuous for x L N 9) A) I only B) All of these are conditions for applying the integral test. C) III only D) II only 2

15) _ Find the sum of the series Q n = 1 ( - 1) n - 1 n 9 n - 1 by expressing 1 1 + x as a geometric series, differentiating both sides of the resulting equation with respect to x, and replacing x by 1 9 . 15) A) 64 81 B) 100 81 C) 81 100 D) 81 64 Find the Maclaurin series for the given function. 16) cos ( - 2x) 16) A) _ Q n = 0 ( - 1) n 2 2n x 2n (2n)! B) _ - Q n = 0 ( - 1) n 2 2n x 2n (2n)! C) _ - Q n = 0 ( - 1) 2n 2 2n x 2n (2n)! D) _ Q n = 0 ( - 1) 2n 2 2n x 2n (2n)! 17) 1 7 + x 17) A) _ Q n = 0 ( - 1) n x n + 1 7 n B) _ Q n = 1 ( - 1) n x n 7 n + 1 C) _ Q n = 0 ( - 1) n x n 7 n D) _ Q n = 0 ( - 1) n x n 7 n + 1 18) e - 2x 18) A) _ Q n = 1 2 n x n n! B) _ Q n = 1 ( - 1) n 2 n x n n! C) _ Q n = 0 2 n x n n! D) _ Q n = 0 ( - 1) n 2 n x n n! 19) sin 6x x 19) A) _ Q n = 0 ( - 1) 2n + 1 6 2n + 1 x 2n (2n + 1)! B) _ Q n = 0 ( - 1) n 6 2n + 1 x 2n (2n + 1)! C) _ Q n = 1 ( - 1) 2n + 1 6 2n + 1 x 2n (2n + 1)! D) _ Q n = 1 ( - 1) n 6 2n + 1 x 2n (2n + 1)! 4

Answer Key Testname: 212_FALL2023_EXAM3MC 1) C 2) D 3) A 4) C 5) D 6) A 7) C 8) B 9) A 10) C 11) C 12) D 13) D 14) A 15) C 16) A 17) D 18) D 19) B 5