9. Use the limit comparison test to determine whether the series converges. b. 1 (24+3)' 10. Use the ratio test to determine whether the series converges. If the test is inconclusive, then say so. b. C. 11. Determine whether the alternating series converges, and justify your answer. E(-1)# #+1 VA + 1 b. E(-1) Ink 12 Use the ratio test for absolute convergence to determine whether the series converges or diverges. If the test is inconclusive, then say so. E(-1) 2* b. E(-D) C. It 13. Classify the series as absolutely convergent, conditionally convergent, or divergent. B. (-4)" b. Sony Ink Sink C. It 1:Determine whether the series converges or diverges. Clearly indicate how you determined each 00 Z "2 cos(n) "3+4 3n4+1 ":1 n21 Use Limit comparison test to determine convergence or diverges 00 oo 2 : 3 2+5" 5n+6 1+7n 12.21 7121 Determine whether the series is absolutely convergent, conditionally convergent or divergent 00 i (3)" (1)""n2 (arctzmn) R Z W Z: W ":1 n21 n21 60 absolutely. 14. If En=, an converges absolutely, prove that E,=, a, converges. Is statement true if 27=1 an converges conditionally? 15. Determine which of the following infinite series converge: (a) 1 M (n + 1)(2n - 1) n n + 1 IN ( C ) 3k Vm + 1 - Vm iM : iM : iM : IM s i m 3m 5m+ 1 100m + 1 (f ) m m 16. (Limit-comparison test.) Prove the following generalization of Theore and Eno b, are series of positive terms such that7. LIMIT COMPARISON TEST Use the LCT to determine whether the infinite series is convergent. n2 e" +n 1 . vi. E 1 p2n _ n2 1 vii. ii. n + In(n) n2 - vn In(n + 4) viii. n 725/2 iMiM &iMi iil. Vn +1 ix. 1 - cos () ] iv. Vn' + 2n2+1 X. E (1-2-1/1) DO 3n + 5 V . n(n - 1)(n - 2) 1=3 Hints: (i) use br = -2. (x) use by =11.7 EXERCISES 1-38 Test the series for convergence or divergence. n2 - 1 1. > n - 1 n=in' + 1 2. n=In + 1 3. > (-1)" n2 - 1 n2 - 1 n' + 1 4. E (-1)" n2+ 1 en n2n 5. 6. iM : IM : it (1 + n) 3n 7 . 8. (-1) *-1 4 n=2 nInn n= 1 TT 2n 00 9. E (- 1)" (2n)! 10. 1=0