Which of the following statements is true?

Question

Luke Alvarez · Accepted Answer

The third statement is correct(ish)

Let's look at each in turn: If a series conditionally converges, then it must absolutely converge as well. False For example, the harmonic series diverges, but the alternating harmonic series converges. #1+1/2+1/3+1/4+...# diverges #1-1/2+1/3-1/4+...# converges If a sequence #alpha_n# converges, then the series #sum alpha_n# converges False Counterexample: harmonic sequence/series. A sequence which is bounded and monotonic must converge This is true for #RR#, but false for #QQ#. For example, a monotonically increasing sequence of approximations to #sqrt(2)# converges in #RR# but not in #QQ#. Nevertheless, this is probably the answer expected. A geometric series converges provided the common ratio is less than #1# False How about common ratio #-2# ? The correct condition would be that the common ratio has absolute value less than #1#.

Isabella Ackerman · Answer

undefined To provide an accurate answer, I would need the statements to choose from. Please provide the statements, and I'll be happy to help determine which one is true.