Use the ratio test to determine convergence or divergence if possible. ' l , 2 " , 3n 1. l 2. n , 3. 7 Eng "le [3] n-ln . (73)\" , {an}! 1 1 4. 5. 6. nZl '1! "21: \"3n "1 2 l1 7. Which of the following series is/are equivalent to 2%? \"'1 * 2(n+1) . 2n . Zn .. 2 7 . 2n \"- Z. n+2 b- Em '- \"Em d- .le'ml 2' THE 8. The function f has derivatives of all orders for all real numbers x. Given f[3) = 2 , f'(3) = 4, 9(3) = 2, and f'\"(3) = 9. a. Write the third degree Taylor polynomial for f about x = 3 b. Use your part a answer to approximate f(2.5} c. Write the fourth degree Taylor polynomial for g[x) = f(x2 +3} about x = 0 9. Given the series f(x) = 3 15x + 75x2 375x3 + a. Write an expression for f(x) using sigma notation. b. Find the interval of convergence of the geometric series. c. Find the exact value of f[0.1) d. Can an exact value of f(0.3) be found? If it can, find it. If not, explain why not. e. Approximate f[0.1) using the first four terms of the series. 1 10. Given f"[x}:10xi, 1'18}: 50, and f[1) = 30 , find an equation for the curve f(x) For 11-17, find the interval of convergence of the given series. (1)\"x" . .. 13. 2" n \":0 n! 11. 2E] 12. "2;" )Pl+1 X" If (1)nl (X _ 4)n e " t _1 14. 2 ii 1 . ( 1 . evils] 5 Z. 4. 6