Conditionally Convergent Series

A conditionally convergent series is a series that converges but does not converge absolutely. In other words, the series converges, but if the absolute values of its terms are taken, the resulting series diverges.

Convergent Series

A series is said to be convergent if the sequence of its partial sums approaches a finite limit as the number of terms increases. In other words, the sum of the series exists and is a finite number.

Absolutely Convergent Series

A series is said to be absolutely convergent if the series of the absolute values of its terms converges. In other words, if the absolute values of the terms are taken and summed, the resulting series converges.

Explanation

Option B is the correct answer because a conditionally convergent series is a series that converges but does not converge absolutely. This means that the series has a finite sum when the terms are summed, but if the absolute values of the terms are taken and summed, the resulting series diverges.

To better understand this concept, let's consider an example:

The alternating harmonic series is an example of a conditionally convergent series:

1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...

This series converges to a finite sum, which is ln(2) ≈ 0.693. However, if we take the absolute values of the terms and sum them, we get the harmonic series:

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ...

The harmonic series, in contrast to the alternating harmonic series, diverges. This means that even though the alternating harmonic series converges to a finite sum, it does not converge absolutely.

Hence, a conditionally convergent series is convergent but not absolutely convergent, which is why option B is the correct answer.