Consider the alternating series:

\$ \sum_{n=1}^\infty ((-1)^(n+1) / n^3)\$.

Determine whether it converges conditionally or absolutely.

A series is said to converge absolutely if the series of absolute values converges.

A series is said to converge conditionally if it converges but does not converge absolutely.

The alternating series test states that if a series has alternating terms that decrease in absolute value and approach zero, then the series converges.

In this case, we have:

\$ \sum_{n=1}^\infty ((-1)^(n+1) / n^3) \$

The terms of this series are decreasing in absolute value and approach zero. Therefore, by the alternating series test, the series converges.

However, the series of absolute values is:

\$ \sum_{n=1}^\infty (1 / n^3) \$

This is a p-series with p = 3 > 1. Therefore, by the p-series test, the series of absolute values converge.

Since the original series converges but the series of absolute values converges as well, we say that the original series converges conditionally.

This option is correct because, the given series satisfies the conditions of the Alternating Series Test, and it converges conditionally

Converges conditionally

The series converges, alternating between positive and negative terms. It does not diverge

Diverges

This is Incorrect. While the absolute series converges, the original series converges conditionally due to its alternating nature

Converges absolutely

This option is incorrect. Option A which converges conditionally, is the correct characterization of the series

None of the above

Option

A

Can you summarize what you’ve understood in the above steps?