Writing: Solutions should be presented in a balanced form3 combining words and sen tences which explain the line of reasoning, and also precise mathematical expressions, formulas and references justifying the steps you are taking are correct. In general; you must give a precise reason for why a sequence is divergent or convergent. Similarly) if the sequence is converge, please justify to the best of your ability the claimed limit in your answer. If you are using theorems in lecture and in the textbook, make that reference clear. {E.g. specify nameumber of the theorem and section of the book.) Telescoping Series In Exercises 39-44, find a formula for the nth partial sum of the series and use it to determine whether the series converges or diverges. If a series converges, find its sum. 39. n=1 n+ 3 3 40. > 72 n=1 (n + 1)2 00 41. (Invn + 1- Invn) n=1 42. ) (tan (n) - tan (n - 1)) n=1 1 43. E arccos - arccos n=1 n+1 n + 2 00 44. ) (Vn +4 - vn+3 n=\fConvergence or Divergence Which series in Exercises 53?6 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. 54. Z (#5)\" 55. i (1)\": 11:1 2\"- CI] 55. Z {1}\"+'n 11:1 on: HIT 5?. . [_) Z ('05 2 11:1] CI] C05 '11?!" ss. 2 m In Exercises 81-86, find the values of x for which the given geometric series converges. Also, find the sum of the series (as a function of x) for those values of x. 81. ) 2" x 7 n=0 82. (-1)"x 2n n=0 83. (-1)"(x +1) 1=0 1 84. 2 (x - 3)" n=0 85. n=0 00 86. (In z) " n=0pplying the Integral Test se the Integral Test to determine whether the series in Exercises 112 converge or diverge. e sure to check that the conditions of the Integral Test are satisfied. Determining Convergence or Divergence Which of the series in Exercises 1346 converge, and which diverge? Give reasons for your answers- \"When you check an answer, remember that there may be more than one way to determine the series\" convergence or divergence.) Limit Comparison Test In Exercises 9-16, use the Limit Comparison Test to determine whether each series converges or diverges. n - 2 9. n3 - n2+3 (Hint: Limit Comparison with En- (12 2 ) ) 10. n+1 n= n2+ 2 (Hint: Limit Comparison with ) (1/v2)) n (n+ 1) 11. iME (n2 + 1) (n - 1) 12. IM 3 +4nDetermining Convergence or Divergence Which of the series in Exercises 1756 converge, and which diverge? Use any method, and give reasons for your answers. 1?. Z; 2v'+ % 13.2w 19. 2 Sign\" 11:1 CIII 20': 1+cosn 21. 22. 23. 24. CI] R 1'1 25. minus) 26. 2?. 11:3 sing the Ratio Test n Exercises 18, use the Ratio Test to determine whether each series converges absolutely Using the Root Test In Exercises 916, use the Root Test to determine whether each series converges absolutely 01' diverges. CO 11. 2 CO 12. 2 CB 8 13. 2 )2n 'n.=l 1.....n(%) onvergenee of Alternating Series n Exercises 114, determine whether the alternating series converges or diverges. Some of he series do not satisfy the conditions of the Alternating Series Test. Absolute and Conditional Convergence Which of the series in Exercises 1548 converge absolutely, which converge conditionally, and which diverge? Give reasons for your answers.