please # 27, 31, 35, 39, 41, 43

25 to 40 Determine whether the series converges or diverges.

x \/1 + n ' n + 2 25. a 2 +11 26. 2:; (n +1); L 5 + 2n " n + 3" 27. , 28. .21 (1+ "31' 2:. 12+ 2" ' e" + l ' | 29. 30. "=1 me" + 1 "g: ruin: l 3,, g m 3,. i w n=l 11' "=1 11' 1 I 2 1 111! 33. 2 1 + e'" 34. 2 "=1 H "=1 H i l L ! 35. 36. 2 " H=l "i "=1 R " . l ' . . l 37. 2 stn() 38. E stn'() 11:1 '1 u:l I1 ' 1 I '* 1 39. tan 40. "I," "=1 H H ":1 H ' 41 44 Use the sum of the rst 10 terms to approximate the sum of the series. Estimate the error. 41. 2 "=1 I 1 I 5+n5 I ,,=. n 43. "21 5 cos'n 44. "5:" 3" + 4" 45. The meaning of the decimal representation of a number 0.d|d3di. . . (where the digit d. is one of the numbers 0, l. 2 ..... 9) is that 0.d1d2d3d4 . . . = _ + _1 + Show that this series converges for all choices of d.. d; ..... 4-6. For what values ofp does the series 23;; 1/(nP1n n) converge? 47. Show that i + ":2 (1n ")1nlnn 49. 50. 51. 52. 53. diverges. [Hint Use Formula 1.5.10 (11" = em") and the fact that lnx 0 and lim,H.. "a" 9'3 0. then 20,. is divergent. Show that if an > 0 and 2a,. is convergent. then 2 ln(1 + an) is convergent. 1f 2 a" is a convergent series with positive terms. is it true that 2 sin(a..) is also convergent? Prove that if a. 2 0 and Ea. converges then 2 0.5 also COIIVCl'gCS. Let 2 a. and 211,. be series with positive terms. ls each ofthe following statements true or false? If the statement is false. give an example that disproves the statement. (a) If 2 a. and 2b,. are divergent. then 2 a,,b,. is divergent. (b) If E a. converges and E b" diverges, then 2a,, b., diverges. (c) If 2 :2,1 and 2 [7,, are convergent. then 2 a,I .1),r is convergent