12.3 Exercises 1. Draw a picture to show that 11. 12. What can you conclude about the series? 2. Suppose f is a continuous positive decreasing function for 13. x > 1 and an = f(n). By drawing a picture, rank the following three quantities in increasing order: 15. [of ( x) dx a ai 1-1 i- 2 17. 3-8 IIll Use the Integral Test to determine whether the series is convergent or divergent. 19. 1 8 1 8 3. 4. 5. n=1 Vn n=1 3n + 1 21 00 n + 2 6. > 7. _ ne 8. n=1 n=in+ 1 23 n=1 9-24 1Ill Determine whether the series is convergent or divergent. 2 9. S 2 10. _ (n-14 + 3n-1.2) n=In 0.85 n=1ergent. This means that Z a,, is ) If ff(x) dx is divergent, then S" f(x) dx -> co as n -> co because f(x) > 0. But (5) Sn-1 - 0. This implies that S, -> co and so E a,, diverges. 11. 1 + + 8 27 64 125 12. 1 + 21/2 + + + 414 515 13. 5 - 2Vn 5 function for 14. In 3 k the following n=1 n= 3 n - 2 00 15. > 16. > 3n + 2 n=In+ 4 n=in(n + 1) 18 17. n 18. n=in+1 n=in' - 4n +5 he series is 18 19. _ ne-n2 In n 20. - n= 1 n= 1 n 2 n 21. > 22. IN n + 1 n=2 n Inn n=In + 1 + 2 1 24. + 1 n = In' + n n=3 n In n In(In n) or divergent. 25-28 IIII Find the values of p for which the series is convergent. 3n -12) 25. 26. E n=2 n(In n)P n=3 n In n [In(In n)]