Provide an appropriate response.
-$\text { If } \sum a _ { n } \text { converges conditionally, what can be said about } \sum \max \left\{ a _ { n } , \left| a _ { n } \right| \right\} \text { ? }$
A) The series may converge or diverge.
B) The series always converges.
C) The series always diverges.

Question

Quizplus · Accepted Answer

Given a series $\sum a_n$ that converges conditionally, it means that $\sum |a_n|$ diverges since conditional convergence is defined by the original series converging while its absolute series diverges. The expression $\max\{a_n, |a_n|\}$ is always equal to $|a_n|$ because $|a_n|$ is always greater than or equal to $a_n$. Therefore, $\sum \max\{a_n, |a_n|\} = \sum |a_n|$, which diverges by the definition of conditional convergence.

Provide an Appropriate Response A) the Series May Converge or Diverge