Convergence, Divergence and Cauchy Sequences - Question Bank

1. Consider the sequence {x_n} where x_n = 1/n for n odd and x_n = 2/n for n even. What is the limit of this sequence?
A) 0
B) 1
C) 2
D) Does not exist
2. If lim sup (x_n) = L and lim inf (x_n) = M, and L != M, what can be said about the sequence {x_n}?
A) It diverges.
B) It converges to L.
C) It converges to M.
D) It is bounded.
3. What is the limit of the sequence {n! / n^n} as n approaches infinity?
A) 1
B) infinity
C) 0
D) e
4. If a sequence {x_n} is not bounded, can it be a Cauchy sequence?
A) No, because Cauchy sequences are always bounded.
B) Yes, it can be.
C) Only if it oscillates.
D) Only if it converges to infinity.
5. What is the limit of the sequence { (2^n + 3^n) / (3^n - 2^n) } as n approaches infinity?
A) 0
B) 1
C) 2
D) 3
6. If {x_n} converges and {y_n} converges, and x_n <= y_n for all n, then lim (x_n) <= lim (y_n). This is an application of:
A) The Squeeze Theorem (or Sandwich Theorem)
B) The Bolzano-Weierstrass Theorem
C) The Cauchy Convergence Theorem
D) The Monotone Convergence Theorem
7. Consider the sequence {x_n} where x_n = 1/n if n is even and x_n = 1/(n+1) if n is odd. What is its limit?
A) 0
B) 1
C) 1/2
D) Does not exist
8. If a sequence {x_n} converges to L, then for any epsilon > 0, there exists an N such that:
A) x_n is in (L - epsilon, L + epsilon) for all n >= N.
B) x_n is outside (L - epsilon, L + epsilon) for all n >= N.
C) x_n = L for all n >= N.
D) x_n > L + epsilon for all n >= N.
9. What is the supremum of the set of limit points of the sequence {(-1)^n}?
A) 1
B) -1
C) 0
D) infinity
10. Which of the following sequences converges to 0?
A) {n / (n+1)}
B) {1/n}
C) {n}
D) {(-1)^n}
11. If lim (x_n) = L, what is lim (x_{n+k}) for a fixed integer k?
A) L
B) L + k
C) k * L
D) Does not exist
12. Consider the sequence {x_n} with x_n = 1 + 1/2 + ... + 1/n (the harmonic series). Does this sequence converge?
A) No, it diverges to infinity.
B) Yes, it converges to a finite value.
C) Yes, it converges to 0.
D) Yes, it converges to 1.
13. What is the limit of the sequence {1/(n*(-1)^n)}?
A) 0
B) 1
C) -1
D) Does not exist
14. If a sequence {x_n} is Cauchy, is it necessarily bounded?
A) Yes
B) No
C) Only if it converges
D) Only if it is monotonic
15. What is the limit of the sequence { (3n^2 + 2n + 1) / (n^2 - n + 5) } as n approaches infinity?
A) 0
B) 1
C) 3
D) infinity
16. If a sequence {x_n} is strictly increasing and unbounded above, what is its limit?
A) Infinity
B) 0
C) 1
D) Does not exist
17. What is the definition of a limit point (or cluster point) of a sequence {x_n}?
A) A value 'a' such that every neighborhood of 'a' contains infinitely many terms of the sequence.
B) A value 'a' such that every neighborhood of 'a' contains at least one term of the sequence.
C) The limit of the sequence if it exists.
D) The supremum of the sequence.
18. Consider the sequence {x_n} where x_n = n*(-1)^n. Does this sequence converge?
A) No, it diverges.
B) Yes, it converges to 0.
C) Yes, it converges to 1.
D) Yes, it converges to -1.
19. If {x_n} is a sequence such that |x_n| -> 0, what can be said about lim (x_n)?
A) lim (x_n) = 0
B) lim (x_n) = 1
C) lim (x_n) does not exist
D) lim (x_n) can be any real number
20. What is the limit inferior (liminf) of the sequence {(-1)^n}?
A) -1
B) 1
C) 0
D) Does not exist
21. In R, if a sequence {x_n} is Cauchy, is it necessarily convergent?
A) Yes, because R is a complete metric space.
B) No, it might diverge.
C) Only if it is monotonic.
D) Only if it is bounded.
22. What is the limit of the sequence { (n^2 + 1) / (2n^2 + n) } as n approaches infinity?
A) 0
B) 1/2
C) 1
D) 2
23. Which of the following sequences is monotonic?
A) {n^2}
B) {(-1)^n}
C) {sin(n)}
D) {cos(n)}
24. If {x_n} converges to L, and {y_n} converges to M, what is lim (x_n - y_n)?
A) L - M
B) L + M
C) L * M
D) L / M (if M != 0)
25. Consider the sequence {x_n} where x_n = n. What is its limit superior?
A) infinity
B) 0
C) 1
D) Does not exist
26. If lim sup (x_n) = lim inf (x_n), what can be said about the sequence {x_n}?
A) It converges to that common value.
B) It diverges.
C) It oscillates.
D) It is unbounded.
27. What is the property of a convergent sequence that allows us to bound its terms?
A) Boundedness
B) Monotonicity
C) Periodicity
D) Oscillation
28. A sequence {x_n} is a Cauchy sequence. Is it guaranteed to converge in the space of rational numbers (Q)?
A) No, for example, the sequence of rational approximations of sqrt(2).
B) Yes, all Cauchy sequences converge in Q.
C) Only if the sequence is monotonic.
D) Only if the sequence is bounded.
29. What is the limit of the sequence { (1 + 1/n)^n } as n approaches infinity?
A) 1
B) e
C) 0
D) infinity
30. If a sequence is monotonic and bounded, what can be concluded about its convergence?
A) It converges.
B) It diverges.
C) It may converge or diverge.
D) It converges to 0.
31. Consider the sequence {x_n} where x_n = 1 if n is odd and x_n = 0 if n is even. Does this sequence converge?
A) No, it oscillates between 0 and 1.
B) Yes, it converges to 0.
C) Yes, it converges to 1.
D) Yes, it converges to 0.5.
32. What is the definition of the limit superior (limsup) of a sequence {x_n}?
A) The largest limit point of the sequence.
B) The smallest limit point of the sequence.
C) The limit of the sequence if it exists.
D) The supremum of the set of limit points.
33. If lim (x_n) = L and lim (y_n) = M, what is lim (x_n * y_n)?
A) L * M
B) L + M
C) L / M (if M != 0)
D) L - M
34. Which of the following sequences is NOT bounded?
A) {1/n}
B) {n/(n+1)}
C) {n}
D) {(-1)^n / n}
35. In the context of real numbers, what is the limit of the sequence {sin(n*pi)}?
A) 0
B) 1
C) -1
D) Does not exist
36. Consider the sequence {n^2}. Does this sequence converge?
A) No, it diverges to infinity.
B) Yes, it converges to 0.
C) Yes, it converges to 1.
D) Yes, it converges to infinity.
37. If a sequence {x_n} converges to L, what is the limit of the sequence {c * x_n} where c is a constant?
A) c * L
B) c + L
C) L / c
D) c
38. Which theorem states that every Cauchy sequence in a complete metric space converges?
A) The Cauchy Convergence Theorem
B) The Bolzano-Weierstrass Theorem
C) The Intermediate Value Theorem
D) The Mean Value Theorem
39. What is the limit of the sequence {2^n / n!} as n approaches infinity?
A) infinity
B) 1
C) 0
D) 2
40. If {x_n} is a convergent sequence, which property must it satisfy?
A) It must be bounded.
B) It must be monotonic.
C) It must be eventually constant.
D) It must be periodic.
41. Consider the sequence {(-1)^n}. Does this sequence converge?
A) No, it oscillates between -1 and 1.
B) Yes, it converges to 0.
C) Yes, it converges to 1.
D) Yes, it converges to -1.
42. In a complete metric space, what is the relationship between convergent sequences and Cauchy sequences?
A) A sequence converges if and only if it is a Cauchy sequence.
B) A sequence converges if it is a Cauchy sequence, but the converse is not always true.
C) A sequence is a Cauchy sequence if it converges, but the converse is not always true.
D) There is no direct relationship between convergence and Cauchy sequences.
43. A sequence {x_n} is called a Cauchy sequence if:
A) For every epsilon > 0, there exists an N such that d(x_n, x_m) < epsilon for all n, m >= N.
B) For every epsilon > 0, there exists an N such that d(x_n, x_m) > epsilon for all n, m >= N.
C) For every epsilon > 0, there exists an N such that d(x_n, x_m) = epsilon for all n, m >= N.
D) For every epsilon > 0, there exists an N such that d(x_n, x_m) is undefined for all n, m >= N.
44. What is the limit of the sequence {n / (n+1)} as n approaches infinity?
A) 0
B) 1/2
C) 1
D) infinity
45. If a sequence {x_n} converges, is it necessarily true that the sequence {|x_n|} also converges?
A) Yes, and to |lim(x_n)|.
B) No, for example, if x_n = (-1)^n.
C) Yes, but to a different limit.
D) Only if x_n is always positive.
46. What does it mean for a sequence {x_n} to be divergent?
A) It does not converge to any limit.
B) It converges to infinity.
C) It oscillates between two values.
D) It is not bounded.
47. Consider the sequence {1/n} for n = 1, 2, 3, ... . What is its limit?
A) 1
B) 0
C) infinity
D) Does not exist
48. Which of the following is a necessary condition for a sequence to converge?
A) It must be bounded.
B) It must be strictly monotonic.
C) It must be periodic.
D) It must be eventually constant.
49. If a sequence {x_n} converges to L, what can be said about the limit of the sequence {x_n + y_n} if {y_n} also converges?
A) It converges to L + lim(y_n).
B) It diverges.
C) It converges to L * lim(y_n).
D) It converges to L / lim(y_n).
50. What is the definition of a convergent sequence in a metric space?
A) A sequence {x_n} converges to x if for every epsilon > 0, there exists an N such that d(x_n, x) < epsilon for all n >= N.
B) A sequence {x_n} converges to x if for every epsilon > 0, there exists an N such that d(x_n, x) > epsilon for all n >= N.
C) A sequence {x_n} converges to x if for every epsilon > 0, there exists an N such that d(x_n, x) = epsilon for all n >= N.
D) A sequence {x_n} converges to x if for every epsilon > 0, there exists an N such that d(x_n, x) is undefined for all n >= N.
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