Question

Which of the following statements is true?

Answer 1

The third statement is correct(ish)

Explanation:

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`.