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

Section 11.4 The Comparison Tests 731
Therefore the remainder R n for the given series satisfies
With n − 100 we have
R 100 <
R n < T n < 1
2n 2
1
2s100d 2 − 0.00005
Using a programmable calculator or a computer, we find that
ò
n−1
1
n 3 1 1 < o 100
n−1
1
n 3 1 1 < 0.6864538
with error less than 0.00005.
n
1. Suppose o a n and o b n are series with positive terms and o b n
is known to be convergent.
(a) If a n . b n for all n, what can you say about o a n? Why?
(b) If a n , b n for all n, what can you say about o a n? Why?
2. Suppose o a n and o b n are series with positive terms and o b n
is known to be divergent.
(a) If a n . b n for all n, what can you say about o a n? Why?
(b) If a n , b n for all n, what can you say about o a n? Why?
3–32 Determine whether the series converges or diverges.
1
3. ò
n−1 n 3 1 8
n 1 1
5. ò
n−1 nsn
7. ò
n−1
4. ò
n−2
9 n
3 1 10 n 8. ò
n−1
1
sn 2 1
n 2 1
6. ò
n−1 n 3 1 1
6 n
5 n 2 1
21. ò
n−1
s1 1 n
2 1 n
5 1 2n
23. ò
n−1 s1 1 n 2 d 2
25. ò
n−1
27. ò
e n 1 1
ne n 1 1
1
n−1S1
nD
1 2
1
29. ò
n−1 n!
31. ò
n−1
sinS 1 nD
e 2n
n 1 2
22. ò
n−3 sn 1 1d 3
24. n 1 3
ò
n
n−1 n 1 2 n
26. ò
n−2
e
28. ò
1yn
n−1 n
n!
30. ò
n−1 n n
1
nsn 2 2 1
1
32. ò
n−1 n 111yn
ln k
9. ò
k−1 k
11. ò
k−1
s 3 k
sk 3 1 4k 1 3
k sin
10. ò
2 k
k−1 1 1 k 3
1 1 cos n
13. ò
14. ò
n−1 e n n−1
s2k 2 1dsk
12. ò
2 2 1d
k−1 sk 1 1dsk 2 1 4d 2
1
s 3 3n 4 1 1
33–36 Use the sum of the first 10 terms to approximate the sum of
the series. Estimate the error.
1
33. ò
n−1 5 1 n 34. e
ò
1yn
5
n−1 n 4
35. ò 5 2n cos 2 n 36. ò
n−1
n−1
1
3 n 1 4 n
4
15. ò
n11
n−1 3 n 2 2
17. ò
n−1
1
sn 2 1 1
n 1 1
19. ò
n−1 n 3 1 n
1
16. ò
n−1 n n
18. ò
n−1
2
sn 1 2
n
20. ò
2 1 n 1 1
n−1 n 4 1 n 2
37. The meaning of the decimal representation of a number
0.d 1d 2d 3 . . . (where the digit d i is one of the numbers 0, 1,
2, . . . , 9) is that
0.d 1d 2d 3d 4 . . . − d1
10 1 d2
10 2 1 d3
10 3 1 d4
10 4 1 ∙ ∙ ∙
Show that this series always converges.
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