Mathematics Questions
Semester III
Unit 3 (Sequences)
1. A sequence [xn ] is said to be _____ sequence, if there exist a real number M1 , such that
xn ≥ M1 ∀ n ∈ N
A Bounded Above
C Bounded
B Bounded below
D Unbounded
Answer: B & C
2. Which of the following statement is/are true?
I. Every convergent sequence is bounded.
II. Every bounded sequence is convergent.
A II is true
C Both are true
B I is true
D None is true
Answer: B
3. If lim |xn | = |l|, then lim xn = l
n→∞
n→∞
A True
C Maybe True
B False
D Can’t Say
Answer: B
4. If the sequence [yn ] and [zn ] converge to 𝑙 then, for Sandwich Theorem to become true (i.e. [xn ]
converges to 𝑙 ) which statement is necessary.
A 𝑦𝑛 < 𝑥𝑛 < 𝑧𝑛
B 𝑦𝑛 > 𝑥𝑛 > 𝑧𝑛
C 𝑦𝑛 < 𝑥𝑛 > 𝑧𝑛
D 𝑦𝑛 = 𝑥𝑛 = 𝑧𝑛
Answer: A
1
1
1
5. The sequence [xn ] = 1 + 3 + 32 + − − − − + 3n , n ∈ N converges to
A
Answer: C
1
2
B
2
3
C
3
D 1
2
3
2
6. A sequence xn is called ____________ if xn+1 ≤ xn ∀ n ∈ N
A Monotonic Increasing Sequence
C Monotonic Decreasing Sequence
B Monotonic Non-Increasing Sequence
D Monotonic Sequence
Answer: B, C and D
1|Rounak Pahuja
Mathematics Questions (Semester III)
Paper I Unit 3 (Sequences)
7. A monotonic ______ sequence is convergent if and only if it is bounded.
A Increasing
C Decreasing
B Non-Increasing
D Non-Decreasing
Answer: A, B, C & D
8. A monotonic decreasing sequence which is not bounded below diverges to
A +∞
C 0
B −∞
D 1
Answer: B
9. Which of the following correctly states the Cauchy General Principal of Convergence?
A A sequence {x_n} converges if and only if it is a Cauchy Sequence.
B A sequence is a Cauchy Sequence if and only if it converges.
C Both A & B
D Neither of A & B
Answer: C
10. Every bounded sequence is a Cauchy sequence.
A True
C True, under certain conditions.
D Can’t say
B False
Answer: True
11. If {xn } & {yn } be two sequences such that lim xn = x and lim yn = y where x and y are finite
n→∞
n→∞
numbers.
Then which of the following is/are incorrect?
A lim (𝑥𝑛 + 𝑦𝑛 ) = 𝑥 + 𝑦
𝑛→∞
B
C
lim (𝑥𝑛 − 𝑦𝑛 ) = 𝑥 − 𝑦
𝑛→∞
lim (𝑥𝑛 + 𝑦𝑛 ) = 𝑥 ∗ 𝑦
𝑛→∞
𝑥
D lim (𝑥𝑛 − 𝑦𝑛 ) = 𝑦 ; given 𝑦 ≠ 0
𝑛→∞
Answer: C & D
12. According to D’Alembert Ratio Test, which of the following is false??
A |l| < 1 → lim xn = 0
C |l| < 1 → lim xn = ∞
B |l| > 1 → lim xn = 0
D |l| > 1 → lim xn = ∞
n→∞
n→∞
n→∞
n→∞
Answer: A & D
2|Rounak Pahuja
Mathematics Questions (Semester III)
Paper I Unit 3 (Sequences)
13. For a given ____ sequence, the limit superior and limit are equal.
A Convergent
C Monotonic
B Divergent
D Cauchy
Answer: A
14. In which condition the sequence {xn } will converge?
A lim sup{xn } = lim inf{xn }
C lim sup{xn } ≠ lim inf{xn }
B lim sup{xn } + lim inf{xn } = 0
D None of the above
Answer: A
15. A sequence {xn } of real numbers converges to x then,
A Sequence {xn2 } converges to x 2
C Sequence {xn2 } converges to x
B Sequence {√x} converges to x 2
D Sequence {xn2 } converges to √x
Answer: A
16. lim
1+3∗10n
n→∞ 5+2∗10n
A
B
= ??
1
C
5
2
3
2
D 1
3
Answer: C
17. lim x n =? ? ; given |x| < 1
n→∞
A 0
B 1
C +∞
D −∞
Answer: A
18. Which of the following statement is/are false regarding Monotonic Increasing Sequence?
A A sequence is convergent if and only if it is bounded.
B A sequence which is not bounded above diverges to −∞
C A sequence which is not bounded below diverges to −∞
D None of the above.
Answer: B
n+1
19. Which of the following are not true regarding the given sequence { n2 } ?
A It is monotonically decreasing sequence.
B It is monotonically increasing sequence.
C It converges to 0
D It is bounded.
3|Rounak Pahuja
Mathematics Questions (Semester III)
Paper I Unit 3 (Sequences)
Answer: B
If not is removed from the question, MSQ question is made with A, C and D correct answers.
1
20. The sequence {n} ∀ n ∈ N is monotonically increasing sequence.
A True
C True, under certain conditions.
B False
D Can’t Say
Answer: B
1
1
1
1
21. If {xn } is a sequence in R where xn = 1 + 2 + 3 + 4 + ⋯ + n , then lim |xn − xn−1 |n =? ?
n→∞
A 0
C Doesn’t Exist
B 1
D ∞
Answer: A
1
3 2 4 3
2
22. lim [(1) (2) (3) . . . . . . . (
n→∞
n+1 n n
n
) ] = ??
A 1
C e
B 1/e
D log10 e
Answer: C
1
23. lim [
(𝑛!)𝑛
𝑛→∞
𝑛
] = ??
A 1/𝑒
C 1
B e
D log10 e
Answer: A
24. lim (𝑥𝑛+1 − 𝑥𝑛 ) = 𝑙 = lim
𝑛→∞
𝑥𝑛
𝑛→∞ 𝑛
A True
B False
Answer: True
25. Identify the correct statement.
A Every bounded sequence is convergent.
B Sequence {𝑥𝑛 } = {−𝑛2 } is divergent to ∞
1
C Sequence {𝑥𝑛 } = {𝑛!} is convergent to 0
D The sequence {𝑥𝑛 } = ((−1)𝑛 ) is convergent to (-1)
Answer: C
4|Rounak Pahuja
Mathematics Questions (Semester III)
Paper I Unit 3 (Sequences)
7𝑛+5
26. The sequence {𝑥𝑛 } = {3𝑛+11} ∀ 𝑛 ∈ 𝑁 converges to
A
B
3
C
7
7
D
3
5
11
11
5
Answer: B
27. Match the following:
2𝑛−7
i.
{𝑥𝑛 } =
ii.
lim (𝑛)𝑛 = 1
3𝑛+2
a. Cauchy’s Sequence
1
𝑛→∞
lim
iii.
𝑛
𝑛→∞ 3𝑛
=0
1
{𝑥𝑛 } = { } , ∀𝑛 ∈ 𝑁
iv.
𝑛
b. Monotonic Increasing Sequence
c. Monotonic Decreasing
Sequence
d. D’Alembert’s Ratio Test
A i-b, ii-a, iii-c, iv-d
C i-a, ii-b, iii-c, iv-d
B i-b, ii-a, iii-d, iv-c
D i-c, ii-d, iii-a, iv-b
Answer:
28. If the sequence {𝑥𝑛 } defined by 𝑥1 = √2 and 𝑥𝑛+1 = √2𝑥𝑛 , then which of the following statement
is/are incorrect?
A The sequence {𝑥𝑛 } is converges to 2.
B The sequence is bounded below by √2.
C It is a monotonic increasing sequence.
D The sequence is bounded above by √2.
Answer: D
1
29. lim 𝑛 sin 𝑛 = ??
𝑛→∞
A 0
B 1
C 2
D -1
Answer: B
5|Rounak Pahuja
Mathematics Questions (Semester III)
Paper I Unit 3 (Sequences)
1 𝑛+1
30. A sequence {𝑥𝑛 } when 𝑥𝑛 = (1 + 𝑛)
∀𝑛 ∈ 𝑁, converges to __
A 𝑒
B −𝑒
C 1/𝑒
D 2𝑒
Answer: A
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