Sequence Convergence: Master the Fundamentals of Calculus

Question 1

Does the sequence defined by a_n = (-1)ⁿ / (n + 1) converge? If so, what is its limit?

Accepted Answer

The sequence converges to 0.

Answer

The sequence diverges to infinity.

Answer

The sequence diverges to negative infinity.

Answer

The sequence converges to 1.

Question 2

Let {xn} be a convergent sequence. Which statement is true about any subsequence of {xn}?

Accepted Answer

It also converges to the same limit as {xn}.

Answer

It may or may not converge to the same limit as {xn}.

Answer

It must converge to a different limit.

Answer

It is impossible to make any conclusions about its convergence.

Question 3

What is a direct consequence of the Bolzano-Weierstrass theorem?

Accepted Answer

Every bounded sequence has a convergent subsequence.

Answer

Every convergent sequence is bounded.

Answer

Every Cauchy sequence is convergent.

Answer

Every divergent sequence is unbounded.

Question 4

Let {xn} be the sequence defined by xn = (-1)^n. As n approaches infinity, what is the limit of this sequence?

Accepted Answer

The limit does not exist.

Answer

0

Answer

1

Answer

-1

Question 5

Which of the following is a sufficient condition for a sequence to be convergent?

Accepted Answer

The sequence is Cauchy.

Answer

The sequence is bounded.

Answer

The sequence is monotone.

Answer

The sequence has a limit point.

Question 6

Let {xn} be the sequence defined by xn = 1/n. As n approaches infinity, what is the limit of this sequence?

Accepted Answer

0

Answer

1

Answer

Infinity

Answer

The limit does not exist.

Question 7

Which property is NOT shared by all convergent sequences?

Accepted Answer

The sequence is bounded.

Answer

The sequence has a unique limit.

Answer

Every subsequence of the sequence converges to the same limit.

Answer

The sequence is Cauchy.

Question 8

Which of the following statements is equivalent to the definition of convergence of a sequence $(x_n)$ to a limit L?

Accepted Answer

For any positive number $epsilon$, there exists a natural number $N$ such that $|x_n - L| < epsilon$ whenever $n > N$.

Answer

For any positive number $epsilon$, there exists a natural number $N$ such that $|x_n - L| > epsilon$ whenever $n > N$.

Answer

For any positive number $epsilon$, there exists a natural number $N$ such that $|x_n - L| = epsilon$ whenever $n > N$.

Question 9

Find the subsequence limit of {n/(n+1)} as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

infinity

Answer

does not exist

Question 10

Use the Bolzano-Weierstrass theorem to prove that the sequence {cos(n)} has a convergent subsequence.

Accepted Answer

The sequence is bounded and closed in R, so it must have a convergent subsequence.

Answer

The sequence is Cauchy, so it must have a convergent subsequence.

Answer

The sequence is monotonic, so it must have a convergent subsequence.

Question 11

Which of the following sequences converges conditionally?

Accepted Answer

{(-1)^n}

Answer

{n}

Answer

{1/n}

Answer

{n^2}

Question 12

Consider the sequence {x_n} where x_n = (n^2 - 1)/(n^2 + 1). Determine the limit of the sequence.

Accepted Answer

1

Answer

does not exist

Answer

infinity

Answer

0

Question 13

Consider the sequence {x_n} where x_n = (-1)^n / (2n + 1). Is {x_n} Cauchy?

Accepted Answer

Yes, by the Cauchy Criterion

Answer

No, by the Divergence Test

Answer

Yes, by the Squeeze Theorem

Answer

No, by the Ratio Test

Question 14

Let {x_n} be a sequence such that x_n ≥ 0 for all n and lim_(n→∞) x_n = 0. Prove that {x_n} is eventually decreasing.

Accepted Answer

Use the definition of convergence and the fact that a nonnegative sequence is decreasing if its terms stop increasing.

Answer

Use the Bolzano-Weierstrass Theorem to find a convergent subsequence.

Question 15

Let {x_n} be a bounded sequence. Prove or disprove: If {x_n} does not converge, then it has a subsequence that diverges to infinity.

Accepted Answer

Prove using the Bolzano-Weierstrass Theorem and the definition of divergence to infinity.

Answer

Prove using the Cauchy Criterion and the definition of convergence.

Answer

Disprove by constructing a bounded sequence that converges to 0 but has no divergent subsequences.

Question 16

Let {x_n} be a sequence of real numbers. Prove that if {x_n} is convergent, then it is bounded.

Accepted Answer

Use the definition of convergence and the triangle inequality to show that the terms of the sequence are eventually within a fixed distance of the limit.

Answer

Use the Bolzano-Weierstrass Theorem to find a convergent subsequence.

Question 17

Consider the sequence {1, 1/2, 1/4, 1/8, ...}. Which of the following statements is correct?

Accepted Answer

It is a convergent sequence with limit 0.

Answer

It is an oscillating sequence with no limit.

Answer

It is a divergent sequence by the Ratio Test.

Answer

It is a Cauchy sequence but not convergent.

Question 18

Which of the following correctly defines convergence of a sequence?

Accepted Answer

As the number of terms approaches infinity, the sequence's terms approach a specific value called the limit.

Answer

The sequence's terms all have the same value.

Answer

The sequence's terms fluctuate randomly without a clear pattern.

Answer

The sequence's terms become increasingly large without bound.

Question 19

Which property is NOT characteristic of a Cauchy sequence?

Accepted Answer

It is always convergent.

Answer

It is bounded.

Answer

For any positive number ε, there exists a natural number N such that |a_m - a_n| < ε for all m, n > N.

Answer

Every convergent sequence is Cauchy.

Question 20

Which of the following statements accurately describes the Bolzano-Weierstrass theorem?

Accepted Answer

Every bounded sequence has at least one convergent subsequence.

Answer

Every Cauchy sequence is bounded.

Answer

Every convergent sequence is Cauchy.

Question 21

Consider the sequence {(-1)^n}. Which of the following is true?

Accepted Answer

It does not converge.

Answer

It converges to 1.

Answer

It converges to 0.

Answer

It converges to -1.

Question 22

Which of the following is NOT a necessary condition for a sequence to converge?

Accepted Answer

It must be monotonic.

Answer

It must be bounded.

Answer

It must have a limit.

Answer

It must be Cauchy.

Question 23

Which of the following sequences converges?

Accepted Answer

{1 / 2^n}

Answer

{n}

Answer

{sin(n)}

Question 24

Consider the sequence {(-1)^n / n}. What is its limit and why?

Accepted Answer

0

Answer

1

Answer

-1

Answer

Does not converge

Question 25

Determine if the sequence {n / (n + 1)} is Cauchy. Provide a justification for your answer.

Accepted Answer

Yes

Answer

Depends on the starting value of n

Answer

No

Answer

Cannot be determined

Question 26

Consider the sequence {a_n} defined by a_n = 1 + 1/2 + 1/4 + ... + 1/2^n. What is the limit of {a_n} and how did you arrive at it?

Accepted Answer

2

Answer

Infinity

Answer

1

Question 27

Which of the following is a necessary and sufficient condition for the convergence of a sequence?

Accepted Answer

The sequence is Cauchy

Answer

The sequence is bounded.

Answer

The sequence has a limit.

Answer

The sequence is monotone.

Question 28

Consider the sequence {a_n} defined by a_n = (-1)^n / n^2. What is the subsequential limit of {a_n}?

Accepted Answer

0

Answer

-1

Answer

1

Question 29

Given a sequence {a_n} that converges to L, what is the limit of the sequence {a_n^2}?

Accepted Answer

L^2

Answer

Does not exist

Answer

0

Answer

L

Question 30

What is the negation of the statement 'The sequence {a_n} converges to L'?

Accepted Answer

There exists an ε > 0 such that for every natural number N, there exists an n > N with |a_n - L| ≥ ε.

Answer

For every ε > 0, there exists a natural number N such that for all n > N, |a_n - L| < ε.

Answer

There exists an ε > 0 such that for all natural numbers N, |a_N - L| ≥ ε.

Answer

For every ε > 0, there exists a natural number N such that for all n > N, |a_n - L| ≥ ε.

Question 31

Which of the following sequences is a Cauchy sequence?

Accepted Answer

a_n = 1/n

Answer

a_n = n

Answer

a_n = (-1)^n

Answer

a_n = sin(n)

Question 32

If {a_n} converges to L and {b_n} converges to M, then the sequence {a_n + b_n} converges to:

Accepted Answer

L + M

Answer

L / M (provided M ≠ 0)

Answer

L * M

Answer

L - M

Question 33

Which of the following is a necessary and sufficient condition for a sequence to converge?

Accepted Answer

It is a Cauchy sequence.

Answer

It has a convergent subsequence.

Answer

It is monotone.

Answer

It is bounded.

Question 34

The Bolzano-Weierstrass theorem states that:

Accepted Answer

Every bounded sequence of real numbers has a convergent subsequence.

Answer

Every sequence of real numbers converges.

Answer

Every bounded sequence of real numbers converges.

Answer

Every convergent sequence of real numbers is bounded.

Question 35

Consider the sequence a_n = (-1)^n / (n + 1). Which of the following statements is true?

Accepted Answer

The sequence converges to 0.

Answer

The sequence converges to 1.

Answer

The sequence diverges.

Answer

The sequence converges to -1.

Question 36

If a subsequence of a sequence converges to L, then:

Accepted Answer

The sequence itself may or may not converge to L.

Answer

The sequence must diverge.

Answer

The sequence must be bounded.

Answer

The sequence must converge to L.

Question 37

Which of the following is an example of a sequence that is bounded but not convergent?

Accepted Answer

a_n = (-1)^n

Answer

a_n = 1/n

Answer

a_n = n/(n+1)

Answer

a_n = (-1)^n/n

Question 38

Which of the following sequences converges to 0?

Accepted Answer

a_n = 1/n

Answer

a_n = (-1)^n

Answer

a_n = n

Question 39

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

Accepted Answer

e

Answer

0

Answer

1

Answer

∞

Question 40

Identify the Cauchy sequence among the following options:

Accepted Answer

a_n = 1/n

Answer

a_n = (-1)^n

Answer

a_n = n

Answer

a_n = n^2

Question 41

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

Accepted Answer

The sequence does not converge.

Answer

The sequence converges to 1.

Answer

The sequence converges to 0.

Answer

The sequence converges to -1.

Question 42

If a sequence (a_n) converges to L, what can be said about its subsequential limits?

Accepted Answer

All subsequential limits must also converge to L.

Answer

Subsequential limits can converge to any value.

Answer

Subsequential limits may not exist.

Answer

All subsequential limits must converge to different values.

Question 43

Given a sequence (a_n) where a_n = n/(n+1), what is the limit of the sequence?

Accepted Answer

The limit is 1.

Answer

The limit is 0.

Answer

The limit is ∞.

Answer

The limit does not exist.

Question 44

Which of the following sequences is NOT monotonic?

Accepted Answer

a_n = (-1)^n * n

Answer

a_n = n^2

Answer

a_n = 1/n

Answer

a_n = n

Question 45

Consider the sequence a_n = sin(nπ/2). Which of the following statements about the sequence is true?

Accepted Answer

The sequence has subsequential limits of 1, -1, and 0.

Answer

The sequence converges to -1.

Answer

The sequence converges to 1.

Question 46

Which condition is NOT sufficient for a sequence to converge?

Accepted Answer

The sequence is bounded.

Answer

The sequence is monotonic and bounded.

Answer

The sequence is Cauchy.

Question 47

Which of the following statements is the negation of the statement 'The sequence {a_n} converges to L'?

Accepted Answer

There exists an ε > 0 such that for every natural number N, there exists an n > N such that |a_n - L| ≥ ε.

Answer

For every ε > 0, there exists a natural number N such that for all n > N, |a_n - L| < ε.

Question 48

A sequence {a_n} is defined recursively by a_1 = 2 and a_(n+1) = (a_n + 3)/2 for n ≥ 1. What is the limit of this sequence?

Accepted Answer

3

Answer

1.5

Answer

2

Answer

The sequence diverges.

Question 49

Which of the following sequences is a Cauchy sequence?

Accepted Answer

{1/n}

Answer

{(-n)^n}

Answer

{(-1)^n}

Answer

{n}

Question 50

If a sequence {a_n} converges to 5 and a sequence {b_n} converges to -2, what is the limit of the sequence {a_n + 2*b_n}?

Accepted Answer

1

Answer

3

Answer

-1

Answer

The sequence diverges.

Question 51

Which of the following is a necessary but not sufficient condition for a sequence to converge?

Accepted Answer

The sequence is bounded.

Answer

The sequence is monotone.

Answer

The sequence is Cauchy.

Answer

The sequence has a unique limit point.

Question 52

Which theorem guarantees that every bounded sequence has a convergent subsequence?

Accepted Answer

Bolzano-Weierstrass Theorem

Answer

Mean Value Theorem

Answer

Intermediate Value Theorem

Answer

Heine-Borel Theorem

Question 53

What is the limit superior of the sequence {(-1)^n + 1/n}?

Accepted Answer

1

Answer

0

Answer

-1

Answer

The limit superior does not exist.

Question 54

If a sequence {a_n} converges to L and f is a function continuous at L, what can be said about the sequence {f(a_n)}?

Accepted Answer

It converges to f(L).

Answer

It may not converge.

Answer

It converges to a value different from f(L).

Answer

It converges to L.

Question 55

Consider the sequence defined by a_n = (n^2 + 1)/(n^2 - 1).  Which technique is most suitable for proving the convergence of this sequence?

Accepted Answer

Divide numerator and denominator by the highest power of n.

Answer

Use the squeeze theorem.

Answer

Apply the monotone convergence theorem.

Question 56

Which of the following statements is FALSE about subsequences?

Accepted Answer

A subsequence of a divergent sequence must also diverge.

Answer

A sequence can have both convergent and divergent subsequences.

Answer

A subsequence of a convergent sequence converges to the same limit as the original sequence.