Sequences: Definition, Limits, and Convergence Tests

Question 1

Determine the limit of the sequence: {a_n} = (n^2 - 1) / (n^2 + 1)

Accepted Answer

1

Answer

The limit does not exist

Answer

-1

Answer

0

Question 2

Determine if the sequence: {a_n} = {(-1)^n} is convergent or divergent.

Accepted Answer

Divergent

Answer

Convergent to 0

Answer

Convergent to 1

Answer

Convergent to -1

Question 3

Consider the sequence: {a_n} = {n^2 - 3n + 2}. Determine the subsequence {a_{2n}}.

Accepted Answer

{2, 6, 12, 20, ...}

Answer

{2}

Answer

{4, 6, 8, 10, ...}

Answer

{1, 0, -1, -4, ...}

Question 4

Determine if the sequence: {a_n} = {cos(nπ/2)} is monotonic.

Accepted Answer

Neither monotonic increasing nor decreasing

Answer

Monotonic increasing

Answer

Monotonic decreasing

Answer

Monotonic but not strictly monotonic

Question 5

Which of the following is a Cauchy sequence?

Accepted Answer

{1, 1/2, 1/4, 1/8, ...}

Answer

{n}

Answer

{(-1)^n}

Answer

{n^2}

Question 6

Determine if the sequence: {a_n} = {n/(n+1)} is bounded.

Accepted Answer

Not bounded

Answer

Bounded above

Answer

Bounded below

Answer

Bounded

Question 7

Determine whether the sequence {(-1)^n} converges or diverges.

Accepted Answer

Diverges

Answer

Converges to -1

Answer

Converges to 1

Answer

Converges to 0

Question 8

Determine whether the sequence {1 + 1 / n} is increasing, decreasing, or constant.

Accepted Answer

Increasing

Answer

Decreasing

Answer

Constant

Answer

Neither increasing nor decreasing

Question 9

Determine whether the sequence (b_n = (-1)^n + 1/n) converges or diverges.

Accepted Answer

Diverges due to oscillations

Answer

Converges to 1

Answer

Converges to 0

Answer

Diverges to infinity

Question 10

Find the limit of the sequence (c_n = sqrt(n^2 + 2n) - n).

Accepted Answer

0

Answer

1

Answer

2

Answer

The limit does not exist

Question 11

Consider the sequence (d_n = 2^n/(3^n + n)). Determine whether this sequence converges or diverges.

Accepted Answer

Converges to 0

Answer

Diverges to infinity

Answer

Converges to 2/3

Answer

Cannot be determined without further information

Question 12

Which convergence test is appropriate for the sequence (e_n = (n^3 - 1)/(n^2 + 2))?

Accepted Answer

Limit comparison test

Answer

Ratio test

Answer

Squeeze theorem

Answer

Alternating series test

Question 13

Find the common ratio of the geometric sequence (2, 6, 18, 54, ...).

Accepted Answer

3

Answer

2

Answer

6

Answer

12

Question 14

Apply the Ratio Test to determine if the sequence $$a_n = rac{n^2+1}{3n^2+2}$$ converges or diverges.

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined using the Ratio Test

Question 15

Identify the type of sequence represented by {1, 1/2, 1/4, 1/8, ...}.

Accepted Answer

Geometric sequence

Answer

Arithmetic sequence

Answer

Fibonacci sequence

Answer

Harmonic sequence

Question 16

Provide a counterexample to the claim that every bounded sequence is convergent.

Accepted Answer

{1, -1, 1, -1, ...}

Answer

{1, 2, 1, 2, ...}

Answer

{1, 1/2, 1/4, 1/8, ...}

Answer

{1, 2, 3, 4, ...}

Question 17

Determine the limit of the sequence {n sin(1/n)} as n approaches infinity.

Accepted Answer

0

Answer

1

Answer

∞

Answer

Does not exist

Question 18

Find the limit of the sequence $$\frac{n^2+3n+2}{n^2+1}$$ as n approaches infinity.

Accepted Answer

1

Answer

2

Answer

3

Answer

4

Question 19

Determine whether the sequence $$\lbrace (-1)^n \rbrace$$ is convergent or divergent.

Accepted Answer

Divergent

Answer

Convergent to 1

Answer

Convergent to -1

Answer

Convergent to 0

Question 20

Which of the following tests can be used to determine the convergence of the sequence $$\frac{n}{n+1}$$?

Accepted Answer

Ratio Test

Answer

Root Test

Answer

Alternating Series Test

Answer

Squeeze Theorem

Question 21

Find the limit of the sequence $$\frac{(n+1)!}{n!}$$ as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

n

Answer

Infinity

Question 22

Consider the sequence $$\lbrace a_n \rbrace$$, where $$a_n = ar^{n-1}$$. What is the condition on r for the sequence to be convergent?

Accepted Answer

$$|r| < 1$$

Answer

$$|r| > 1$$

Answer

$$|r| = 1$$

Answer

No condition on r

Question 23

Determine the convergence of the sequence $$\frac{\ln(n)}{n}$$.

Accepted Answer

Convergent to 0

Answer

Convergent to 1

Answer

Divergent to infinity

Answer

Divergent to -infinity

Question 24

Which of the following sequences is increasing?

Accepted Answer

$$lbrace n^2+1 \rbrace$$

Answer

$$lbrace (-1)^n \rbrace$$

Answer

$$lbrace \sin(n) \rbrace$$

Answer

$$lbrace \frac{1}{n} \rbrace$$

Question 25

Find the limit of the sequence $$\left(1+\frac{1}{n}\right)^n$$ as n approaches infinity.

Accepted Answer

e

Answer

0

Answer

1

Answer

Infinity

Question 26

Let {an} be a sequence defined by an = (-1)^n / n. Which of the following statements correctly describes the behavior of {an}?

Accepted Answer

The sequence oscillates between -1 and 1.

Answer

The sequence diverges to negative infinity.

Answer

The sequence converges to -1.

Answer

The sequence converges to 0.

Question 27

Find the limit of the sequence {an} = n^2 / (n^2 + 1) as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

2

Answer

The limit does not exist.

Question 28

Apply the Ratio Test to determine the convergence or divergence of the series ∑(n=1 to ∞) (n + 1) / n^2.

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined with the Ratio Test

Question 29

Let {an} and {bn} be convergent sequences such that lim(n->∞) an = 2 and lim(n->∞) bn = 3. Find lim(n->∞) (an - bn) / (an + bn).

Accepted Answer

0

Answer

1

Answer

2

Answer

The limit does not exist.

Question 30

Which of the following statements about arithmetic sequences is correct?

Accepted Answer

An arithmetic sequence is a sequence where the difference between any two consecutive terms is constant.

Answer

An arithmetic sequence is a sequence where the ratio between any two consecutive terms is constant.

Answer

An arithmetic sequence is a sequence where the sum of any two consecutive terms is constant.

Answer

An arithmetic sequence is a sequence where the product of any two consecutive terms is constant.

Question 31

Which of the following sequences is geometric?

Accepted Answer

{an} = 3^n

Answer

{an} = 2n + 1

Answer

{an} = (-1)^n

Answer

{an} = n^2 + 1

Question 32

Find the general term of the sequence given a_1 = 2 and a_n = 3a_{n-1} + 1.

Accepted Answer

a_n = 2 * (3^(n-1))

Answer

a_n = 2 * (2^(n-1))

Answer

a_n = 3 * (3^(n-1))

Answer

a_n = 3 * (2^(n-1))

Question 33

Given the sequence (a_n = (-1)^n), does a convergent subsequence exist?

Accepted Answer

Yes, the subsequence of even-indexed terms converges to 1.

Accepted Answer

Yes, the subsequence of odd-indexed terms converges to -1.

Answer

No, the sequence oscillates between -1 and 1.

Answer

Yes, the sequence is bounded.

Question 34

Determine the limit of the sequence {n/(n+1)} as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

Infinity

Question 35

Write the correct notation for the limit of the sequence {a_n} as n approaches infinity.

Accepted Answer

lim (n→∞) a_n

Answer

lim∞→n a_n

Answer

inf lim a_n

Question 36

Determine whether the sequence (a_n = \frac{cos(n\pi)}{n}) is bounded.

Accepted Answer

Bounded

Answer

Unbounded

Answer

None of the above

Answer

Cannot be determined

Question 37

Which of the following sequences converges to 0 as n approaches infinity?

Accepted Answer

a_n = 1/n^2

Answer

a_n = n

Answer

a_n = 2^n

Answer

a_n = (-1)^n

Question 38

Find the limit of the sequence a_n = (n^2 + 1)/(n^2 - 1) as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

∞

Answer

-1

Question 39

The sequence a_n = (1 + 1/n)^n converges to:

Accepted Answer

e

Answer

1

Answer

0

Answer

∞

Question 40

If a sequence is both bounded and monotonic, what can you conclude about its convergence?

Accepted Answer

Convergent

Answer

Divergent

Answer

Oscillating

Answer

Unbounded

Question 41

The sequence defined recursively by a_1 = 1, a_n = a_(n-1) + 2 for n ≥ 2 is:

Accepted Answer

Arithmetic

Answer

Harmonic

Answer

Geometric

Answer

Fibonacci

Question 42

Find the limit of the sequence a_n = (n + 1) / (n - 1) as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

-1

Answer

∞

Question 43

Which of the following sequences converges to 0 as *n* approaches infinity?

Accepted Answer

a_n = (1/n)ⁿ

Answer

a_n = n² + 1

Answer

a_n = 2n - 1

Answer

a_n = n/(n+1)

Question 44

Find the limit of the sequence a_n = √(n² + 1) - n as *n* approaches infinity.

Accepted Answer

0

Answer

1

Answer

Does not exist

Answer

Infinity

Question 45

What is the general term (a_n) for the arithmetic sequence with a first term (a₁) of 5 and a common difference (d) of -2?

Accepted Answer

a_n = 5 - 2(n-1)

Answer

a_n = 5 + 2(n-1)

Answer

a_n = 5 + 2n

Answer

a_n = 5n - 2

Question 46

Is the sequence a_n = 1/n convergent or divergent?

Accepted Answer

Convergent

Answer

Divergent

Answer

Cannot be determined

Answer

Oscillatory

Question 47

Which test is commonly used to determine the convergence of a sequence whose terms are positive and strictly decreasing?

Accepted Answer

Monotone Convergence Theorem

Answer

Ratio Test

Answer

Limit Comparison Test

Answer

Root Test

Question 48

If the sequence a_n = (n³ + 2)/(n² + 1) converges, what is its limit as *n* approaches infinity?

Accepted Answer

Infinity

Answer

1

Answer

2

Answer

0

Question 49

Consider the sequence a_n = (-1)ⁿ. Which of the following statements is true about this sequence?

Accepted Answer

The sequence oscillates between -1 and 1.

Answer

The sequence converges to 1.

Answer

The sequence converges to 0.

Answer

The sequence diverges to infinity.

Question 50

Which of the following sequences is a geometric sequence?

Accepted Answer

an = 2n

Answer

an = n!

Answer

an = 1/n

Answer

a_n = n² + 1

Question 51

What is the sum of the first 10 terms of the geometric sequence with a first term (a₁) of 2 and a common ratio (r) of 3?

Accepted Answer

59048

Answer

1944

Answer

243

Answer

972

Question 52

Which of the following is the formula for the nth term (a_n) of the Fibonacci sequence?

Accepted Answer

a_n = (1/√5)*((1+√5)ⁿ - (1-√5)ⁿ)

Answer

a_n = n! + 1

Answer

a_n = n² - 1

Answer

a_n = 2ⁿ + 1