Convergence Tests: Divergence, Integral, and p-Series 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: • Recognize series that cannot converge by applying the Divergence Test. • Use the Integral Test on appropriate series (all terms positive, corresponding function is decreasing and continuous) to make a conclusion about the convergence of the series. • Recognize a p-series and use the value of p to make a conclusion about the convergence of the series. • Use the algebraic properties of series. PRACTICE PROBLEMS: For problems 1 { 9, apply the Divergence Test. What, if any, conlcusions can you draw about the series? 1 X 1. (−1)k k=1 lim (−1)k DNE and thus is not 0, so by the Divergence Test the series diverges. k!1 Also, recall that this series is a geometric series with ratio r = −1, which confirms that it must diverge. 1 X 1 2. (−1)k k k=1 1 lim (−1)k = 0, so the Divergence Test is inconclusive. k!1 k 1 X ln k 3. k k=3 ln k lim = 0, so the Divergence Test is inconclusive. k!1 k 1 1 X ln 6k 4. ln 2k k=1 ln 6k lim = 1 6= 0 [see Sequences problem #26], so by the Divergence Test the series diverges. k!1 ln 2k 1 X 5. ke−k k=1 lim ke−k = 0 [see Limits at Infinity Review problem #6], so the Divergence Test is k!1 inconclusive. 1 X ek − e−k 6. ek + e−k k=1 ek − e−k lim = 1 6= 0 [see Sequences problem #21], k!1 ek + e−k so by the Divergence Test the series diverges. 1 k X 1 7. 1 + k k=1 1k lim 1 + = e 6= 0 [see Sequences problem #34], k!1 k so by the Divergence Test the series diverges. 1 X p 8. ( k2 + 8k − 5 − k) k=1 p lim ( k2 + 8k − 5 − k) = 4 6= 0 [see Sequences problem #28], k!1 so by the Divergence Test the series diverges. 1 X p p 9. ( k2 + 3 − k2 − 4) k=2 p p lim ( k2 + 3 − k2 − 4) = 0, so the Divergence Test is inconclusive.; Detailed Solution: Here k!1 For problems 10 { 20, determine if the series converges or diverges by applying the Divergence Test, Integral Test, or noting that the series is a p-series. Explic- itly state what test you are using. If you use the Integral Test, you must first verify that the test is applicable. If the series is a p-series, state the value of p. 2 1 X ln k 10. k k=3 The series diverges by the Integral Test. 1 X 11. ke−k k=1 The series converges by the Integral Test. 1 X 1 12. arctan − arctan(k) k k=1 The series diverges by the Divergence Test. [see Sequences problem #33.] 1 X 1 p 13. 4 k=1 k + 15 1 1 X 1 X 1 1 p = p , which is a p-series with p = < 1, so the series diverges. 4 4 4 k=1 k + 15 k=16 k 1 X 14. πke−k k=1 The series diverges by the Divergence Test. Also, observe that this is a geometric series π with ratio r = e > 1, which confirms that the series diverges. 1 X 1 15. 4k2 k=2 The series is a constant multiple of a p-series with p = 2 > 1, so the series converges. 1 X k2 16. 4k2 + 9 k=2 The series diverges by the Divergence Test. 1 X k 17. 4k2 + 9 k=2 The series diverges by the Integral Test. 3 1 X 1 18. 4k2 + 9 k=2 The series converges by the Integral Test.; Detailed Solution: Here 1 X 1 19. 4k2 − 9 k=2 The series converges by the Integral Test.; Detailed Solution: Here 1 X 20. 15k−0:999 k=10 The series is a constant multiple of a p-series with p = 0:999 < 1, so the series diverges. For problems 21 & 22, use algebraic properties of series to find the sum of the series. 1 X 1 1 1 21. − − 6k k k + 1 k=1 4 − 5 1 1 4 1 8 1 16 22. + 2 − + + + − + + ::: 2 4 7 8 49 16 343 [Hint: See Infinite Series problems #11 & #12.] 47 15 4.