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Exam 2 Practice Problems - Calculus II | MATH 231, Exams of Calculus

University of Illinois - Urbana-ChampaignCalculus

Material Type: Exam; Class: Calculus II; Subject: Mathematics; University: University of Illinois - Urbana-Champaign; Term: Summer 2008;

Typology: Exams

Pre 2010

Uploaded on 03/11/2009

koofers-user-42c
koofers-user-42c 🇺🇸

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Answers to the practice problems for Exam 2
Math 231 A1, Summer 2008
19. 8, by the geometric series formula.
21. 4/3, by the geometric series formula.
22. 12/5, by the geometric series formula.
25. diverges, by the kth-term test or the integral test.
26. diverges, by the kth-term test.
27. converges, by the alternating series test.
28. diverges, by the integral test or the limit comparison test with P1/k1/2.
29. diverges, since it is 3 times a p-series with p < 1.
30. converges, since it is 2 times a p-series with p > 1.
31. converges, by the converges by comparison (or limit comparison) with P1/k5/2.
32. diverges, by the limit comparison test with P1/k1/2.
33. converges, by the alternating series test or the ratio test.
34. diverges, by the kth-term test or the ratio test or the root test.
35. converges, by the alternating series test.
36. converges, by noticing that coskπ = (−1)kand then using the alternating series test.
37. converges, by the integral test, or the comparison test or limit comparison test with P2/k2.
38. diverges, by the integral test.
39. diverges, by the kth-term test or the ratio test.
40. converges, by the ratio test or root test.
41. converges, by the integral test, or the comparison test or limit comparison test with P1/k2.
42. diverges, by the integral test.
43. converges, by the ratio test.
44. diverges, by the limit comparison test with P1/k.
45. converges (by the alternating series test), but only conditionally (use the integral test or the limit
comparison test with P1/k when testing for absolute convergence).
46. converges (by the alternating series test), but only conditionally (use the integral test or the limit
comparison test with P1/k when testing for absolute convergence).
47. converges absolutely by comparison with P1/k3/2
48. converges (by the alternating series test), but only conditionally (use the comparison test with P3/k
when testing for absolute convergence).
49. converges for p > 1 (by the integral test or the limit comparison test for P1/kp).
pf3

Partial preview of the text

Download Exam 2 Practice Problems - Calculus II | MATH 231 and more Exams Calculus in PDF only on Docsity!

Answers to the practice problems for Exam 2

Math 231 A1, Summer 2008

19. 8, by the geometric series formula.

21. 4/3, by the geometric series formula.

22. 12/5, by the geometric series formula.

25. diverges, by the kth-term test or the integral test.

26. diverges, by the kth-term test.

27. converges, by the alternating series test.

28. diverges, by the integral test or the limit comparison test with

1 /k

29. diverges, since it is 3 times a p-series with p < 1.

30. converges, since it is 2 times a p-series with p > 1.

31. converges, by the converges by comparison (or limit comparison) with

1 /k

32. diverges, by the limit comparison test with

1 /k^1 /^2.

33. converges, by the alternating series test or the ratio test.

34. diverges, by the kth-term test or the ratio test or the root test.

35. converges, by the alternating series test.

36. converges, by noticing that cos kπ = (−1)

k

and then using the alternating series test.

37. converges, by the integral test, or the comparison test or limit comparison test with

2 /k

38. diverges, by the integral test.

39. diverges, by the kth-term test or the ratio test.

40. converges, by the ratio test or root test.

41. converges, by the integral test, or the comparison test or limit comparison test with

1 /k^2.

42. diverges, by the integral test.

43. converges, by the ratio test.

44. diverges, by the limit comparison test with

1 /k.

45. converges (by the alternating series test), but only conditionally (use the integral test or the limit

comparison test with

1 /k when testing for absolute convergence).

46. converges (by the alternating series test), but only conditionally (use the integral test or the limit

comparison test with

1 /k when testing for absolute convergence).

47. converges absolutely by comparison with

1 /k^3 /^2

48. converges (by the alternating series test), but only conditionally (use the comparison test with

3 /k

when testing for absolute convergence).

49. converges for p > 1 (by the integral test or the limit comparison test for

1 /kp).

50. converges when |e

p

| < 1, or in other words p < 0. The series behave as described below. Following

each series is a list of tests that could successfully be applied; it’s possible there are other tests which

would work.

Here are the answers for A through X:

n=

sin^2 n

∑^3 n^ converges (comparison test with the geometric series

n

n=

n+

n diverges (limit comparison test with^

n ).

n=

(−1)n

n! converges absolutely (ratio test).

n=

n n

diverges (comparison test with

n

, or limit comparison test with the p-series

n^2 /^3

, or

integral test).

n=

n

ln n

diverges (integral test).

n=

2 n

n

converges (comparison with the geometric series

(1/2)n, or ratio test, or root test).

n=

(−1)n 23 n 7 n^

diverges (nth-term test for divergence, or ratio test, or root test).

n=

nn^

converges (comparison test with

n

or

n^2

, or ratio test, or root test).

n=

[3+(−1)n]n^ converges (comparison test with^

n

n=1 n

)n

converges (ratio test, or root test).

n=

n

2 n+1 diverges (nth term test for divergence, or integral test, or comparison test with^

n ).

n=

n

n^4 +2n^2 +1 converges (integral test, or comparison test/limit comparison test with^

n^3 ).

n=

n^2 − 3 converges (limit comparison test with^

n^2 ).

n=

cos nπ

n^3 /^2 converges absolutely (|(cos^ nπ)/n

| = 1/n

, which is a p-series with p > 1).

n=

√^1

2 n^3 −n

converges (limit comparison test with

n^3 /^2

n=

n 4 n^2 +

diverges (limit comparison test with

n

, or integral test).

n=

3 n+ 5 en+n

diverges (limit comparison test with the geometric series

(3/e)n/5, or ratio test).

n=

4 n− 1

diverges (limit comparison test with

n

, or integral test).

n=

n 1+n

diverges (limit comparison test with

n^1 /^2

n=

n!

1 · 3 · 5 ···(2n−1) converges (ratio test).

n=

n^3

n! converges (ratio test).

n=

(n!)^2

(2n)! diverges (ratio test).

n=

n

(n+1)(n+3)(n+5) converges (comparison test, or limit comparison test, with^

n^2 ).