Convergence Tests: Divergence, Integral, and P-Series Tests

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?

∞ X 1. (−1)k k=1 lim (−1)k DNE and thus is not 0, so by the Divergence Test the series diverges. k→∞ Also, recall that this series is a geometric series with ratio r = −1, which conﬁrms that it must diverge.

∞ X 1 2. (−1)k k k=1 1 lim (−1)k = 0, so the Divergence Test is inconclusive. k→∞ k

∞ X ln k 3. k k=3 ln k lim = 0, so the Divergence Test is inconclusive. k→∞ k

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→∞ ln 2k

∞ X 5. ke−k k=1 lim ke−k = 0 [see Limits at Inﬁnity Review problem #6], so the Divergence Test is k→∞ inconclusive.

∞ X ek − e−k 6. ek + e−k k=1 ek − e−k lim = 1 6= 0 [see Sequences problem #21], k→∞ ek + e−k so by the Divergence Test the series diverges.

∞ k X 1 7. 1 + k k=1 1k lim 1 + = e 6= 0 [see Sequences problem #34], k→∞ k so by the Divergence Test the series diverges.

∞ X √ 8. ( k2 + 8k − 5 − k) k=1 √ lim ( k2 + 8k − 5 − k) = 4 6= 0 [see Sequences problem #28], k→∞ so by the Divergence Test the series diverges.

∞ X √ √ 9. ( k2 + 3 − k2 − 4) k=2 √ √ lim ( k2 + 3 − k2 − 4) = 0, so the Divergence Test is inconclusive.; Detailed Solution: Here k→∞

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 ∞ X ln k 10. k k=3 The series diverges by the Integral Test.

∞ X 11. ke−k k=1 The series converges by the Integral Test.

∞ X 1 12. arctan − arctan(k) k k=1 The series diverges by the Divergence Test. [see Sequences problem #33.]

∞ X 1 √ 13. 4 k=1 k + 15 ∞ ∞ X 1 X 1 1 √ = √ , which is a p-series with p = < 1, so the series diverges. 4 4 4 k=1 k + 15 k=16 k

∞ 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.

∞ X 1 15. 4k2 k=2 The series is a constant multiple of a p-series with p = 2 > 1, so the series converges.

∞ X k2 16. 4k2 + 9 k=2 The series diverges by the Divergence Test.

∞ X k 17. 4k2 + 9 k=2 The series diverges by the Integral Test.

3 ∞ X 1 18. 4k2 + 9 k=2 The series converges by the Integral Test.; Detailed Solution: Here

∞ X 1 19. 4k2 − 9 k=2 The series converges by the Integral Test.; Detailed Solution: Here

∞ 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.

∞ 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