Is it possible for a series of positive terms to converge conditionally? Explain.

Verified step by step guidance

1. Recall the definitions: A series \( \sum a_n \) converges absolutely if \( \sum |a_n| \) converges, and it converges conditionally if \( \sum a_n \) converges but \( \sum |a_n| \) diverges.
2. Note that for a series of positive terms, each term \( a_n > 0 \), so \( |a_n| = a_n \). This means the series and its absolute value series are the same.
3. Since the series and its absolute value series are identical for positive terms, if the series converges, it must converge absolutely.
4. Therefore, a series with positive terms cannot converge conditionally because conditional convergence requires the absolute value series to diverge.
5. In summary, conditional convergence is only possible when the series has terms that are not all positive (i.e., some terms are negative or alternate in sign).

Key Concepts

Positive Term Series

A series with positive terms means each term is greater than zero. Such series are always non-negative and cannot have terms that cancel each other out, which affects the type of convergence they can exhibit.

Absolute Convergence

A series converges absolutely if the series of the absolute values of its terms converges. For positive term series, absolute convergence is equivalent to regular convergence since all terms are positive.

Conditional Convergence

Conditional convergence occurs when a series converges, but the series of absolute values diverges. This typically requires terms of varying signs, so a series with only positive terms cannot converge conditionally.