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