Part A [15 Points]: Choose TRUE or FALSE for each of the following items.
- Given that lim?→∞
?? =
5
3
and lim?→∞
?? = −2 , we have lim?→∞
(3?? − 4??
) = −3
TRUE FALSE - Given that {??
} = {(
−?−3
7?+5
)
2
}, we have lim?→∞
?? =
1
49
.
TRUE FALSE - The series ∑
3?
−4?+1
∞
?=1
is convergent.
TRUE FALSE - The series ∑
(−5)
?−1
2
?
∞
?=1
converges with sum 1
7
.
TRUE FALSE - The series ∑ ?
−4/7
is divergent using the ?-Series Test.
TRUE FALSE
Part B [25 Points]: Chose the correct answer for each of the following items. - Which of the following is equal to lim?→∞
−3+4?
3
2−?3
?
A) −
3
2
B) −4
C) −∞ D) +∞ - The sequence ?? = (−
3
7
)
?
, as ? → ∞ , is
A) Divergent B) Convergent to 3
7
C) Convergent to −
3
7
D) Convergent to 0 - The sum of the geometric series 7 −
7
2
+
7
4
−
7
8
- ⋯ is equal to
A) 2
3
B) 14
C) 14
3
D) −14
- For the alternating series ∑ (−1)
?−1?
−4
3 ∞ ⁄
?=1 which one gives lim?→∞
???
A) lim?→∞
?? = 0 B) lim?→∞
?? = 1
C) lim?→∞
?? = ∞ D) None of them - For the alternating series 3
10
−
7
19
+
11
28
−
15
37
- ⋯ which one is the formula for ???
A) ?? =
4?
9?
B) ?? =
?+2
?+9
C) ?? =
4?−1
9?+1
D) None of them
Part C [60 Points]: Solve the following problems. Show your work clearly.
- Use the ?-Series Test to determine whether each of the following series is convergent or
divergent.
Hint: In the Lecture Note 3, see Exercises 3 & 4 on Slide 20.
?) ∑
1
√?
−3
∞
?=1
?) ∑
3
?
7
5
⁄
∞
?=1
?) ∑
3?
6
7?
5
∞
?=1 - What is the Integral Test in general? Use the Integral Test to determine whether the
series is convergent or divergent.
∑
5
√3? + 1
∞
?=1
Hint: In the Lecture Note 3, see Example 4 on Slide 17 and Exercise 6 on Slide 21. - Write the nth term of the following series. Determine whether the series converges or
diverges.
∑
1
5?
2 + 8
∞
?=1
Hint: In the Lecture Note 3, see Exercise 17 on Slide 35. - What is the Alternating Series Test? Use it to test the following series for convergence or
divergence:
∑
(−1)
?
√9? − 2
∞
?=1
Sample Solution