Mathematics

This problem asks you to classify a series with alternating signs by testing two things separately: whether it converges in absolute value (absolute convergence), and whether the original alternating series converges on its own (conditional convergence). Start by checking if $\sum \frac{5}{n^{3/2}}$ converges using the $p$-test, then verify that the alternating series satisfies the conditions for the alternating series test. The key insight is that absolute convergence is the strongest form and will immediately settle the question.

This problem asks you to classify a series with alternating signs by testing two things separately: whether it converges in absolute value (absolute convergence), and whether the original alternating series converges on its own (conditional convergence). Start by checking if $\sum \frac{5}{n^{3/2}}$ converges using the $p$-test, then verify that the alternating series satisfies the conditions for the alternating series test. The key insight is that absolute convergence is the strongest form and will immediately settle the question.