James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

CHAPTER 11 Review 785
18. If a n . 0 and lim n l ` sa n11ya nd , 1, then lim n l ` a n − 0.
19. 0.99999 . . . − 1
20. If lim an − 2, then lim san13 2 and − 0.
n l ` n l `
21. If a finite number of terms are added to a convergent series,
then the new series is still convergent.
22. If ò a n − A and ò b n − B, then ò a nb n − AB.
n−1
n−1
n−1
EXERCISES
;
1–8 Determine whether the sequence is convergent or divergent.
If it is convergent, find its limit.
1. a n − 2 1 n3
9n11
2. an −
3
1 1 2n 10 n
3. a n − n3
4. an − cossny2d
2
1 1 n
5. a n − n sin n
n 2 1 1
6. a n − ln n
sn
7. hs1 1 3ynd 4n j 8. hs210d n yn!j
9. A sequence is defined recursively by the equations a 1 − 1,
a n11 − 1 3 san 1 4d. Show that hanj is increasing and an , 2
for all n. Deduce that ha nj is convergent and find its limit.
10. Show that lim n l ` n 4 e 2n − 0 and use a graph to find the
smallest value of N that corresponds to « − 0.1 in the precise
definition of a limit.
11–22 Determine whether the series is convergent or divergent.
n
11. ò
n−1 n 3 1 1
n
12. ò
2 1 1
n−1 n 3 1 1
n
13. ò
3
n−1 5 14. ò
n
n−1
15. ò
n−2
17. ò
n−1
1
nsln n
16. ò
n−1
cos 3n
1 1 s1.2d n 18. ò
n−1
1 ? 3 ? 5 ? ∙ ∙ ∙ ? s2n 2 1d
19. ò
n−1 5 n n!
s25d
20. ò
2n
n−1 n 2 9 n
sn
21. ò s21d n21
n−1 n 1 1
sn 1 1 2 sn 2 1
22. ò
n−1 n
s21d n
sn 1 1
lnS n
3n 1 1D
n 2n
s1 1 2n 2 d n
23–26 Determine whether the series is conditionally convergent,
absolutely convergent, or divergent.
23. ò s21d n21 n 21y3
n−1
24. ò s21d n21 n 23
n−1
s21d
25. ò
n sn 1 1d3 n
s21d
26. ò
n sn
n−1 2 2n11 n−2 ln n
27–31 Find the sum of the series.
s23d
27. ò
n21
28. ò
n−1 2 3n n−1
29. ò ftan 21 sn 1 1d 2 tan 21 ng
n−1
31. 1 2 e 1 e 2
2! 2 e 3
3! 1 e 4
4! 2 ∙ ∙ ∙
1
nsn 1 3d
s21d
30. ò
n n
n−0 3 2n s2nd!
32. Express the repeating decimal 4.17326326326 . . . as a
fraction.
33. Show that cosh x > 1 1 1 2 x 2 for all x.
34. For what values of x does the series oǹ−1 sln xdn converge?
s21d
35. Find the sum of the series ò
n11
correct to four deci mal
n−1 n
places.
5
36. (a) Find the partial sum s 5 of the series oǹ−1 1yn6 and estimate
the error in using it as an approximation to the sum
of the series.
(b) Find the sum of this series correct to five decimal places.
37. Use the sum of the first eight terms to approximate the sum of
the series oǹ−1 s2 1 5n d 21 . Estimate the error involved in this
approximation.
n
38. (a) Show that the series ò
n
is convergent.
n−1 s2nd!
(b) Deduce that lim
n l `
n n
s2nd! − 0.
39. Prove that if the series oǹ−1 an is absolutely convergent, then
the series
n−1S ò
n 1 1
Da n
n
is also absolutely convergent.
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