The Comparison, Limit Comparison, Ratio, & Root Tests

SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference your lecture notes and the relevant chapters in a textbook/online resource. EXPECTED SKILLS: • Use the following tests to make a conclusion about the convergence of series with no negative terms: - Comparison Test - Limit Comparison Test - Ratio Test - Root Test PRACTICE PROBLEMS: For problems 1 & 2, apply the Comparison Test to determine if the series converges. Clearly state to which other series you are comparing.

∞ X 1 1. 3k + 5 k=1 1 1 < for k ≥ 1. 3k + 5 3k ∞ ∞ X 1 X 1 Since converges (geometric series, r = 1 ), must converge. 3k 3 3k + 5 k=1 k=1

∞ X 1 2. 3(k − 2) k=3 1 1 1 = > for k ≥ 3. 3(k − 2) 3k − 6 3k ∞ ∞ X 1 X 1 Since diverges (p-series with p = 1), must diverge as well. 3k 3(k − 2) k=3 k=3

For problems 3 & 4, apply the Limit Comparison Test to determine if the series converges. Clearly state to which other series you are comparing.

1 ∞ X 1 3. 3(k + 2) k=1 1 k 1 lim 3(k+2) = lim = , which is ﬁnite and nonzero. k→∞ 1 k→∞ k 3k + 6 3 ∞ ∞ X 1 X 1 So since diverges (Harmonic Series), must diverge as well. k 3(k + 2) k=1 k=1

∞ X 1 4. 3k − 5 k=2

1 k k 3 lim 3 −5 = lim = 1, which is ﬁnite and nonzero. k→∞ 1 k→∞ k 3k 3 − 5 ∞ ∞ X 1 X 1 So since converges (geometric series, r = 1 ), must converge. 3k 3 3k − 5 k=2 k=2 For problems 5 – 7, apply the Ratio Test to determine if the series converges. If the Ratio Test is inconclusive, apply a diﬀerent test. ∞ X 1 5. k! k=0 The series converges by the Ratio Test.

∞ X 1 6. 3k k=0 The series converges by the Ratio Test. [Of course this just conﬁrms what we already 1 knew as this is a geometric series with r = 3 .]

∞ X 1 7. 3(k + 2) k=1 The Ratio Test is inconclusive; however, as shown in #3 above, the series diverges by the Limit Comparison Test. For problems 8 – 10, apply the Root Test to determine if the series converges. If the Root Test is inconclusive, apply a diﬀerent test. ∞ X 1 8. 3k k=0 The series converges by the Root Test. [Of course this just conﬁrms what we already 1 knew as this is a geometric series with r = 3 .]

2 ∞ k X 1 9. 1 + k k=1 The Root Test is inconclusive; however, as shown in the previous assigned problems, Convergence Tests #7, the series diverges by the Divergence Test.

∞ X k 10. 7k k=1 The series converges by the Root Test.; Detailed Solution: Here For problems 11 – 22, apply the Comparison Test, Limit Comparison Test, Ratio Test, or Root Test to determine if the series converges. State which test you are using, and if you use a comparison test, state to which other series you are comparing to. ∞ X 1 √ 11. 3 k=1001 k − 10 ∞ X 1 √ The series diverges by the Comparison Test. Compared to 3 . k=1001 k

∞ X 4k2 + 5k 12. √ 5 k=1 10 + k ∞ X 1 The series diverges by the Limit Comparison Test. Compared to √ .; k=1 k Detailed Solution: Here

∞ X 2k + 1 13. (2k)! k=0 The series converges by the Ratio Test.

∞ X k2 cos2 k 14. 2 + k5 k=1 ∞ X 1 The series converges by the Comparison Test. Compared to . k3 k=1

∞ X 1 15. k(5k) k=1 The series converges by the Root Test.

3 ∞ X 2 + 2k 16. 3 + 3k k=0

∞ k X 2 The series converges by the Limit Comparison Test. Compared to . 3 k=0

∞ X 6(k+1) 17. k! k=0 The series converges by the Ratio Test.

∞ X 6k + k 18. [Hint:Use the result from the previous problem.] k! + 6 k=0 ∞ X 6(k+1) The series converges by the Comparison Test. Compared to .; Detailed Solution: Here k! k=0

∞ X k3 − 2 19. 2 2 k=2 (k + 1) ∞ X 1 The series diverges by the Limit Comparison Test. Compared to . k k=2

∞ X arctan k 20. k1.5 k=1 ∞ X π/2 The series converges by the Comparison Test. Compared to . k1.5 k=1

∞ X 7k 21. k2 + | sin k| k=1 ∞ X 1 The series diverges by the Limit Comparison Test. Compared to . k k=1

∞ X (2k)! 22. (k!)2 k=0 The series diverges by the Ratio Test.