Quiz on Sequences and Limits

What is a convergent sequence?

• A sequence that approaches a finite limit

• A sequence that diverges to infinity

• A sequence that oscillates indefinitely

• A sequence with no defined limit

What characterizes a divergent sequence?

• It approaches a specific value

• It does not approach any limit

• It is always increasing

• It is always decreasing

What is the limit superior of a sequence?

• The average of all terms in the sequence

• The smallest limit point of the sequence

• The largest limit point of the sequence

• The first term of the sequence

What is the limit inferior of a sequence?

• The smallest limit point of the sequence

• The largest limit point of the sequence

• The median of the sequence

• The last term of the sequence

What defines a Cauchy sequence?

• The terms become arbitrarily close to each other

• The terms diverge to infinity

• The sequence is strictly increasing

• The sequence is strictly decreasing

Which of the following is true about convergent sequences?

• They can be divergent

• They are always Cauchy sequences

• They have no limit

• They oscillate indefinitely

What happens to a Cauchy sequence in a complete metric space?

• It diverges

• It oscillates

• It becomes constant

• It converges

Which of the following is an example of a divergent sequence?

• 1/n as n approaches infinity

• (-1)^n

• 1 + 1/n as n approaches infinity

• n as n approaches infinity

What is the relationship between limit superior and limit inferior?

• They are always equal

• Limit superior is always less than limit inferior

• They are independent of each other

• Limit superior is always greater than or equal to limit inferior

Which of the following sequences is Cauchy?

• (-1)^n

• n

• n^2

• 1/n