Please #3,7,11,15,19,23,27,31,39,41

1. Draw a picture to show that 8 n 7. > 8. Ene-" n=2 n + 1 1=1 18 15 dx 00 tan n 9. > 10. n=2 n(In n)3 What can you conclude about the series? 2. Suppose f is a continuous positive decreasing function for 11-28 Determine whether the series is convergent or divergent. x > 1 and an = f(n). By drawing a picture, rank the follow- ing three quantities in increasing order: 11. 12. En-0.9999 n=1 nv2 1=3 [of( x) dx I'M ai 13. 1 + + + + 8 27 64 125 3-10 Use the Integral Test to determine whether the series is 14. + + + + convergent or divergent. 7 11 13 3. En-3 4. En-03 15. - + + + + n=1 3 7 11 15 19 2 5. 6. 16. 1 + + + + + iM n=1 5n - 1 (3n - 1)4 21/2 4 5 15Use Euler's result to find the sum of the series. 17. In + 4 Vn iM : 18. ( b) E- Is ( k - 2 ) + S IM : 19. 20. n=1 n' + 2n + 2 38. (a) Find the partial sum $10 of the series En- 1*. Estimate 21. 22. 3n - 4 the error in using $10 as an approximation to the sum n=in* + 4 1-3 n2 - 2n of the series. In n (b) Use (3) with n = 10 to give an improved estimate of the 23. M : 24. n=2 n nn 2 12 sum. (c) Compare your estimate in part (b) with the exact value 25. > ke-k 26. > ke-42 given in Exercise 37. k=1 (d) Find a value of n so that S,, is within 0.00001 of the sum. 39. (a) Use the sum of the first 10 terms to estimate the sum of IM : 27. n in + n' 28. in' + 1 the series En=1 1'. How good is this estimate? (b) Improve this estimate using (3) with n = 10. (c) Compare your estimate in part (b) with the exact value 29-30 Explain why the Integral Test can't be used to determine given in Exercise 36. whether the series is convergent. (d) Find a value of n that will ensure that the error in the approximation s * S, is less than 0.001. 29. COS Trn 30. \\ cos'n Vn 40. Find the sum of the series 2,=1 ne-2" correct to four decimal places. 31-34 Find the values of p for which the series is convergent. 41. Estimate E,=1 (2n + 1)- correct to five decimal places. 31. 42. How many terms of the series ,-21/[n(In n)'] would you 32. > n=2 n(In n)P "=3 n In n [In(In n)]P need to add to find its sum to within 0.01? 43. Show that if we want to approximate the sum of the series 33. E n(1 + n2) 34. In n n=1 np En=In 100 so that the error is less than 5 in the ninth decimal place, then we need to add more than 1011, terms! 35-37 The Riemann Zeta Function The function & defined by T 44. (a) Show that the series E,=1 (In n) ' is convergent. (b) Find an upper bound for the error in the approxima- 1 tion s Sn. 5 (s ) = 2- (c) What is the smallest value of n such that this upper bound is less than 0.05? where s is a complex number, is called the Riemann zeta function. (d) Find s, for this value of n. 35. For which real numbers x is { (x) defined? 45. (a) Use (4) to show that if s,, is the nth partial sum of the harmonic series, then 36. Leonhard Euler was able to calculate the exact sum of the p-series with p = 2: Sn 1 + Inn (b) The harmonic series diverges, but very slowly. Use 5(2) = E- part (a) to show that the sum of the first million terms is less than 15 and the sum of the first billion terms is less Use this fact to find the sum of each series. than 22. (a) > - 46. Use the following steps to show that the sequence 2 n2 ( b ) > 1 =2 thi = 1 + - - In n (c) "= (2n)2 has a limit. (The value of the limit is denoted by y and is 37. Euler also found the sum of the p-series with p = 4: called Euler's constant.) (a) Draw a picture like Figure 6 with f(x) = 1/x and 5 (4) = > 1 interpret t, as an area [or use (5)] to show that t, > 0 for 90 all n