( b) Ap to show that of k. Explain why we cannot use a p-series with 0 K, where K is an arbitrary positive integer. 15. Fill in the blanks and select the correct word: Explain why your statement is valid. If lim uk = 0 and Ek=1 - k- co bk _converges, then Ex 6. Explain how you could adapt the limit comparison test (converges/diverges). to analyze a series Ck=1 ak in which all of the terms are negative. K-+0o bk 16. If lim 2k = 0 and Ek=1 ak converges, explain why v 7. Provide a more general statement of the limit comparison cannot draw any conclusions about the behavior test in which Ex-1 ak and Ek=1 bk are two series whose Ek=1 bk. terms are eventually positive. Explain why your statement 17. Fill in the blanks and select the correct word: is valid. If lim uk = co and Ek=1 diverges, then Ek=1 8. If you suspect that a series _k=1 ak diverges, explain why (converges/diverges). you would need to compare the series with a divergent series, using either the comparison test or the limit com- K-+ 0o bk 18. If lim uk = co and Ek_1 ak diverges, explain why parison test. cannot draw any conclusions about the behavior 9. If you suspect that a series _=1ak converges, explain why you would want to compare the series with a conver- 19. In Example 1 we used the comparison test to show gent series, using either the comparison test or the limit 700 4 3 /2 - K - 1 the series _k=1 5k3 + 3 converges. Use the limit co comparison test. parison test to prove the same result. 10. Briefly outline the advantages and disadvantages of using the two comparison tests to analyze the behavior 20. In Example 1 of Section 7.4 we used the integral test of a series _=1 ak. show that the series _=1 2k2 converges. Use the li comparison test with the series Ck=1 x to prove same result. Skills In Exercises 21-30 use one of the comparison tests to de- termine whether the series converges or diverges. Explain 1 + Ink 8 27. K2 28. how the given series satisfies the hypotheses of the test you K= 1 ( k - IT ) 2 use. 1 + Ink 29 ( sink) 21 3/2+ 1 30. 3 k 3 K * 3 + * 2 + 5 22. K= 1 k= 1 KEO K K = 1 (1 + 2k) 2 Use any convergence test from this section or the previous 23, 3k -5 VK tion to determine whether the series in Exercises 31-48 24. K = 2 kv k 3 - 4 1+ k 2 verge or diverge. Explain how the series meets the hypoth K = 1 of the test you select. 1+Ink 1 31. K- 1/2 32. K- 4/3 K= 1 K 26. M k + 0.01 K = 1 K= 1 K = 1