) If {an} and {bn} are divergent, then {an bn} is divergent. b) If {an} and {bn} are divergent, then {an, bn} is divergent. c) If {an} is convergent and {bn} is divergent, then {an bn} is divergent. d) If {an} is convergent, then {an} is convergent.
The Correct Answer and Explanation is:
Correct Answer: c) If {an} is convergent and {bn} is divergent, then {an bn} is divergent.
A sequence is a list of numbers in a specific order, and a sequence is called convergent if its terms approach a fixed value as the index increases. A sequence is divergent if it does not approach a finite value.
Statement c asserts that if one sequence, {an}, is convergent and the other, {bn}, is divergent, then the product sequence {an bn} must be divergent. This statement is true, and this can be confirmed by the definition of convergence and divergence.
To illustrate, suppose {an} converges to a finite number L, and {bn} does not settle on any finite value. The product {an bn} will behave erratically if {bn} grows without bound or oscillates, and therefore, {an bn} will also fail to approach a single finite number. For example, if {an} converges to 1 and {bn} is the sequence {n}, then {an bn} = {n}, which diverges to infinity. If {bn} = (–1)^n, an oscillating divergent sequence, then even if {an} = 1, the product remains oscillating and hence divergent.
Statement a is false because two divergent sequences can have a product that converges. Consider {an} = (–1)^n and {bn} = (–1)^n; both diverge, but their product {an bn} = 1 for all n, which converges.
Statement b is vague, as {an, bn} suggests combining the sequences, perhaps as a tuple or interleaving them, which does not lead to a clear convergence analysis without more definition.
Statement d is trivial and always true, but also circular in logic. Saying a convergent sequence is convergent does not add new information.
Thus, option c is the best and most accurate logical statement.
