Can you please help me with 30.

592 CHAPTER 8 INFINITE SEQUENCES AND SERIES 38-39 Let 27. 28. Vn lim Va. | = L 10 30. sin 4n The Root Test says the following: 29 2 1 (n + 1)42 +1 (i) If Z 1 (or L = 0), then E a, is divergent. (- 1)" arctan n (-2)"n! 31. (iii) If L = 1, then the Root Test is inconclusive. n (2n)! (Like the Ratio Test, the Root Test is proved by comparison with a 33. 1 - - 1 . 3 1 . 3 . 5 1 . 3 . 5 . 7 . . . geometric series.) Determine whether the given series is absolutely 3! 5! 7! convergent. + (-1)"-1 1 . 3 . 5 . . . . . (2n - 1) - + .. . 2n - 1)! 38. E ( 2")" 34. 2 2 . 6 2 . 6 . 10 2 . 6 . 10 . 14 5 . 8 5 . 8 . 11 5 . 8 . 11 . 14 40. For which positive integers & is the following series convergent? 35. The terms of a series are defined recursively by the equations at = 2 Sn + 1 -1 (kn)! An + 3 41. (a) Show that E-o x"! converges for all x. Determine whether E a, converges or diverges. (b) Deduce that lim, -- x"! = 0 for all x. 36. A series _ a, is defined by the equations 42. Around 1910, the Indian mathematician Srinivasa Ramanujan 2 + cos " discovered the formula Vn Determine whether E a, converges or diverges. 1 2V2 (4n)!(1103 + 26390n) 9801- 37. For which of the following series is the Ratio Test inconclusive (n!) 3964 (that is, it fails to give a definite answer)? William Gosper used this series in 1985 to compute the first (a) b) 17 million digits of 7. (a) Verify that the series is convergent. (-3)"-1 (b) How many correct decimal places of i do you get if you (c) (d) 51 +n use just the first term of the series? What if you use two Vn terms? EC 8.5 Power Series on Converse / diverge A power series is a series of the form 1 [cax" = co+ cix + cax? + cax' + ... where x is a variable and the C,'s are constants called the coefficients of the series. For each fixed x, the series (1) is a series of constants that we can test for convergence or divergence. A power series may converge for some values of x and diverge for other values of x. The sum of the series is a function f ( x) = co+ cix + czx? + . . . + cax"+ ... whose domain is the set of all x for which the series converges. Notice that f resembles a polynomial. The only difference is that f has infinitely many terms. PDF Scanned with MOBILE SCANNER