MATH 104 Practice Problems for Exam 3

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1 MATH 4 Practice Problems for Exam 3 There are too many problems here for one exam, but they re good practice! For each of the following series, say whether it converges or diverges, and explain why n 3 n n 4 n + n 4 n 2 n n sin n 5. How many terms of the series are needed to get a partial sum that is within n3. of the actual sum of the series? Give the smallest possible number of terms. 6. Does 2 n + 3 n 4 n converge? If so, what is the sum? 7. For what values of p does the series n p 2 + n 3 converge? 8. Find the precise interval of convergence of the series (2x 5) n n 2 4 n. 9. Which of the following is the beginning of the Maclaurin series for arctan (x 2 )? (A) x 2 x4 3 + x6 5 x8 x (B) 7 3 x6 6 + x 9 x (C) x 2 2x 4 + 3x 6 4x (D) + x 4 + 3x 8 + 4x (E) x2 3 + x6 6 + x 9 + x (F) x2 x6 3 + x 5 x

2 x. Let F (x) = series for F? cos t dt. Which of the following is the beginning of the Maclaurin (A) x 2 + x2 24 x3 x (B) x x3 72 x (C) x x2 2 + x4 6 x6 x (D) x4 24 x (E) x x2 3 + x4 5 x5 x (F) x x4 5 + x Use the first two non-zero terms of an appropriate series to give an approximation of sin(x 2 ) dx. Give (with explanation) an estimate of the error (difference between your approximation and the actual value of the integral). 2. We would like to estimate the sum of the series, by using the sum of the n first ten terms. Of course, the exact error is the sum of all the terms from the th on, i.e.,. Show that this error is less than /3 by comparing this n with the sum of /n 4 and then by estimating this latter sum using an appropriate integral. 3. Find the center and radius of convergence of the power series n=2 ( ) n + n n 2 (x 2) n. (In other words, find the largest open interval on which the series converges). 4. Write the second-degree Taylor polynomial for f(x) = x centered at a = 25. Use this polynomial from to estimate 26. Also, give an estimate of the error. 5. Determine whether the series converges or diverges. Please explain carefully. n n n 2 6. Does the sequence {(n 2 + n) /n } converge or diverge? If it converges, calculate its limit.

3 7. Let a n = n + (x 2 + x 4 ) dx. Does the sequence {a n} converge? If so, what is 2 the limit? 8. Determine whether the following series converges: ln( + n=3 n ) 2 9. Does the series 2. Find the limit. n!(n + )! (3n)! converge or diverge? Be sure to justify your answer. lim x cos x e x2 The rest of these are problems from exams the last few times I ve given the course. { ( )} n 2 2. What is the limit of the sequence arctan? + n 2 (a) (b) (c) π 4 (d) ln 3 (e) ln 9 (f) diverges 22. Does the series n=2 n 2 n 23. What is the limit of the sequence { cos ( )} n? + n (a) (b) (c) π 4 (d) ln 3 (e) ln 9 (f) diverges 24. Does the series n=2 n n

4 25. Does the series n! n5 26. What is the limit of the sequence 3 (a) (b) (c) Does the series 28. Does the series 29. Does the series 3. Does the series n= n + n ( {n 2 cos )}? n (d) 2 nn 3 n n3 3 (e) 2 2n + 3 (n 2 + 3n + 6) 2 (f) diverges (x 4) n 3. For which values of x does the series converge? What is the sum? n= 5 n (a) < x <, (b) < x <, (c) < x < 9, x 4 9 x x (d) < x < 9, (e) < x <, (f) < x < 9, 9 x 5 x 5 x 32. Does the series ( ) n 2 n 3 n + 4n absolutely (b) Converges conditionally (c) Diverges

5 33. Does the series n 5 n! 34. Does the series diverge? ( ) n (n 2 + n 3 ) n 4 + converge absolutely, converge conditionally, or absolutely (b) Converges conditionally (c) Diverges 35. Does the series (2n)! 2 n (n!) Does the series 2 n (n!) 2 (2n)! 37. Does the series ( ) n n ln n absolutely (b) Converges conditionally (c) Diverges 38. For which values of p does the series n= (n 2 + ) p converge? (a) p > (b) p < (c) p > 2 (d) p < ln 5 ln 3 (e) p < ln 5 ln For which values of x does the series ( ) n (x 2) n converge? n3 n (f) p < (a) < x 5 (b) < x < 5 (c) x 5 (d) 5/3 x 7/3 (e) 3 < x < 3 (f) 3 x 3

6 4. For which values of p does the series ln( + n p ) converge? n= (a) p > (b) p < (c) p > (d) p < ln 5 ln 3 (e) p < ln 5 2 ln 3 (x 2) n 4. For which values of x does the series converge? n 2 3 n (f) p < (a) < x 5 (b) < x < 5 (c) x 5 (d) 5/3 x 7/3 (e) 3 < x < 3 (f) 3 x For which values of p does the series n= 3 np 4 n + 5 n converge? (a) p > (b) p < (c) p > (d) p < ln 5 ln 3 (e) p < ln 5 2 ln 3 3 n (x 2) n 43. For which values of x does the series converge? n 2 (f) p < (a) < x 5 (b) < x < 5 (c) x 5 (d) 5/3 x 7/3 (e) 3 < x < 3 (f) 3 x What are the first few nonzero terms of the Maclaurin series for f(x) = xe 3x? (a) 2x x x6 + (b) x + 4 x2 + 8 x3 + 2 x4 + (c) x x3 + 6 x x5 + (d) + 2x x x6 + (e) x 2 4 x3 + 6 x4 64 x5 + (f) x + 3x x x Which of the following gives the value of within /)? (a) 2 (b) 2 2 (c) /2 (d) 2 32 e x5 dx correct to within. (i.e., (e) (f) What are the first few nonzero terms of the Maclaurin series for f(x) = 4x2 4 x? (a) 2x x x6 + (b) x + 4 x2 + 8 x3 + 2 x4 +

7 (c) x x3 + 6 x x5 + (d) + 2x x x6 + (e) x 2 4 x3 + 6 x4 64 x5 + (f) x + 3x x x Which of the following gives the value of (i.e., within /)? (a) 2 (b) 2 2 (c) /2 (d) 2 32 cos(x 2 ) dx correct to within. (e) (f) What are the first few nonzero terms of the Maclaurin series for f(x) = cos(2x)? (a) 2x x x6 + (b) x + 4 x2 + 8 x3 + 2 x4 + (c) x x3 + 6 x x5 + (d) + 2x x x6 + (e) x 2 4 x3 + 6 x4 64 x5 + (f) x + 3x x x Which of the following gives the value of (i.e., within /)? (a) 2 (b) 2 2 (c) /2 arctan(x 2 ) dx correct to within. (d) 2 32 (e) (f) MATH 4-4 Third Midterm Exam - Fall 2. What is the limit of the sequence {sin(arctan(ln(n)))}?. What is the limit of the sequence {cos(arctan(ln(n)))}?. What is the limit of the sequence {ln(cos(arctan(n)))}? (a) (b) (c) π 4 (d) ln 2 (e) π 2 (f) diverges 2. Does the series n=2 n n 3 n +

8 2. Does the series 2. Does the series n=2 n=2 n 2 n 3 n + n n 2 n + 3. Does the series 3. Does the series 3. Does the series 2 n n5 n 5 2n n 5 5 n n! 4. Does the series ( ) n (n 2 + n) n 3 + 2n + 4. Does the series diverge? 4. Does the series diverge? ( ) n (n 2 + n 3 ) n 5 + ( ) n (n 2 + n 3 ) n 4 + ln n converge absolutely, converge conditionally, or converge absolutely, converge conditionally, or absolutely (b) Converges conditionally (c) Diverges 5. If it converges, find the sum of the series why. ( ) n+ n2 n If the series diverges, explain

9 5. If it converges, find the sum of the series why. n= ( ) n π 2n 9 n (2n)! If the series diverges, explain 5. If it converges, find the sum of the series n= ( ) n If the series diverges, explain why. n! 2n (a) ln 2 (b) ln 3 ln 2 (c) / e (d) /2 (e) /e 2 (f) Diverges 6. Does the series n=2 ( ) n ln(n!) 6. Does the series n=2 ( ) n n 3 + ln(n!) 6. Does the series n=2 ( ) n ln(n!) n5 + absolutely (b) Converges conditionally (c) Diverges 7. For which values of p does the series 7. For which values of p does the series 7. For which values of p does the series + n p converge? + np converge? n + n p converge? (a) p > (b) p < (c) p > 2 (d) p > 2 (e) p < (f) no value of p 8. For which values of x does the series 8. For which values of x does the series (x ) n converge? n 2 (x ) n + 2 n converge?

10 8. For which values of x does the series nx n 2 n converge? (a) < x < 3 (b) < x < 2 (c) x 3 (d) x 2 (e) 2 < x < 2 (f) 2 x 2 9. What are the first few nonzero terms of the Maclaurin series for f(x) = x cos(2x)? 9. What are the first few nonzero terms of the Maclaurin series for f(x) = 2 sin(2x)? 9. What are the first few nonzero terms of the Maclaurin series for f(x) = xe 2x2? (a) 2x x x6 + (b) x + 4 x2 + 8 x3 + 2 x4 + (c) x x3 + 6 x x5 + (d) x 2x x x7 + (e) x 2 3 x x x7 + (f) x 2x 3 + 2x x7 +. Which of the following is closest to the value of answer.. + x2 dx? Justify your. Which of the following is closest to the value of. Which of the following is closest to the value of.. e x2 /2 dx? Justify your answer. cos(x 2 ) dx? Justify your answer. (a).998 (b).999 (c). (d). (e).2 (f).3

11 MATH 4 Third Midterm Exam - Fall 24. What is the limit of the sequence defined recursively by a = and a n+ = a n + 6?. What is the limit of the sequence defined recursively by a = and a n+ = a n + 2? (a) 3 (b) 4 (c) 5 (d) 6 (e) 2 (f) 2. What is the limit of the sequence defined recursively by a = and a n+ = a n + 2? (a) 3 (b) 4 (c) 5 (d) 6 (e) 2 (f) 2 2. Does the series 2. Does the series 2. Does the series n + 3 n 2 n + 5 n + 2 n n + 4 n + 5 n 3 n + 3 n 2 3 n 3. Does the series n! 3. Does the series 3. Does the series n! n 2 3n 2 n n3

12 4. Does the series 4. Does the series 4. Does the series ( ) tan n 2 ( ) sin n 2 ( ) cos n 2 5. If it converges, find the sum of the series explain why. n= ( ) n π 2n+ 6 2n+ (2n + )! If the series diverges, 5. If it converges, find the sum of the series why. 5. If it converges, find the sum of the series why. ( ) n+ n3 n ( ) n 2 n n= n! If the series diverges, explain If the series diverges, explain (a) ln 3 (b) ln 4 ln 3 (c) e 2 (d) /2 (e) /e 2 (f) Diverges 6. Does the series ( ) n n 2 n Does the series ( ) n (n 2 + ) n 3 6. Does the series ( ) n n 2 n 4 + absolutely (b) Converges conditionally (c) Diverges

13 7. For which values of p does the series 7. For which values of p does the series 7. For which values of p does the series n ( + n 2 ) p converge? ( + n 3 ) p converge? n 2p ( + n 2 ) p converge? (a) p > (b) p < (c) p > 3 (d) p > 3 (e) p < (f) no value of p 8. For which values of x does the series 8. For which values of x does the series (x 2) n n 2 2 n (x 2) n n + converge? converge? 8. For which values of x does the series x n n 2 2 n converge? (a) 2 < x 2 (b) < x < 4 (c) x 4 (d) x 3 (e) x < 3 (f) 2 x 2 9. What are the first few nonzero terms of the Maclaurin series for f(x) = arctan(x 2 )? 9. What are the first few nonzero terms of the Maclaurin series for f(x) = sin(x 2 )? 9. What are the first few nonzero terms of the Maclaurin series for f(x) = x 2 cos(x 2 )? (a) 2x x4 + (b) x 2 2 x x + (c) x x6 + 6 x + (d) x 2 3 x6 + 5 x + (e) x 2 6 x6 + 2 x + (f) x 2 6 x6 + 5 x +

14 .. Which of the following is closest to the value of sin x dx? Justify your answer.. Which of the following is closest to the value of answer.. arctan(x) dx? Justify your. Which of the following is closest to the value of. xe x2 dx? Justify your answer. (a).4975 (b).498 (c).4986 (d).4992 (e).4996 (f).4999 MATH 4 Third Midterm Exam - Fall 25. For each of the following series, say whether it converges or diverges and which test you would use to prove it. (You don t have to give the details of applying the test) (.) n n Test: (.2) 2 n n! Test: (.3) ln n Test: (.4) (n!) 2 Test:

15 2. What is the limit of the sequence defined recursively by a = and a n+ = 2 + a n? 2. What is the limit of the sequence defined recursively by a = and a n+ = 4 + a n? 2. What is the limit of the sequence defined recursively by a = and a n+ = 8 + a n? (a) + 2 (b) + 3 (c) (d) (e) (f) Does the series (n + ) 3/2 n 3 n Does the series n + 2 n n Does the series (n + 3) 3/2 n 2 n Does the series (2n)! (n!) 2 4. Does the series (n!) 2 (2n)! 4. Does the series 2 n n! (2n)!

16 ( 5. Does the series e /n ) 5. Does the series 5. Does the series ( e /n e /n) ( e /n 2) 6. If it converges, find the sum of the series why. n= ( ) n π 2n 3 2n (2n)! If the series diverges, explain 6. If it converges, find the sum of the series why. ( ) n+ n2 n If the series diverges, explain 6. If it converges, find the sum of the series n= ( ) n 2 n! If the series diverges, explain why. (a) ln 2 (b) ln 3 ln 2 (c) /e 2 (d) /2 (e) 2/e (f) Diverges 7. Does the series ( ) n n 2 ln n n Does the series diverge? ( ) n (n 2 + ln n) n 3 converge absolutely, converge conditionally, or 7. Does the series ( ) n n 2 ln n n 4 + absolutely (b) Converges conditionally (c) Diverges

17 8. For which values of p does the series 8. For which values of p does the series 8. For which values of p does the series n p + n 2 converge? ( + n p ) 4 converge? n p ( + n 2 ) p converge? (a) p < (b) p > (c) p > 4 (d) p > 4 (e) p < (f) no value of p ( ) n+ (x ) n 9. For which values of x does the series n 4 n converge? 9. For which values of x does the series (x + ) n (n + ) 4 n converge? ( ) n+ (x + ) n 9. For which values of x does the series n 2 4 n converge? (a) 3 < x < 5 (b) 3 x < 5 (c) 3 < x 5 (d) 5 < x 3 (e) 5 x < 3 (f) 5 x 3. Using the Taylor series for 3 x centered at x = 8, determine which of the following is closest to the value of Justify your answer.. Using the Taylor series for 4 x centered at x = 6, determine which of the following is closest to the value of Justify your answer.. Using the Taylor series for 5 x centered at x = 32, determine which of the following is closest to the value of Justify your answer. (a) 2.4 (b) 2.5 (c) 2.6 (d) 2.7 (e) 2.8 (f) 2.9

18 .2. Which of the following is closest to the value of cos(x 2 ) dx? Justify your answer.. Which of the following is closest to the value of.2 e 2x2 dx? Justify your answer.. Which of the following is closest to the value of.2 dx? Justify your answer. + x2 (a).95 (b).96 (c).97 (d).98 (e).99 (f).2

MATH 104 Practice Problems for Exam 3

MATH 104 Practice Problems for Exam 3 MATH 14 Practice Problems for Exam 3 There are too many problems here for one exam, but they re good practice! For each of the following series, say whether it converges or diverges, and explain why. 1..