The alternating harmonic series ∑ (-1)^n / n converges. Does it converge absolutely, conditionally, or diverge?

The alternating harmonic series ∑ (-1)^n / n converges. Does it converge absolutely, conditionally, or diverge?

• A. Diverges
• B. Converges absolutely
• C. Oscillates
• D. Converges conditionally

Answer: (D)

The main idea is distinguishing types of convergence for a series. This alternating harmonic series has terms that alternate in sign and decrease in magnitude to zero, so it passes the Alternating Series Test and converges. But if you take absolute values, you get the harmonic series ∑ 1/n, which diverges. Since the series itself converges but its absolute value series diverges, the convergence is conditional, not absolute. (If you’re curious, the sum equals -ln 2.)

The main idea is distinguishing types of convergence for a series. This alternating harmonic series has terms that alternate in sign and decrease in magnitude to zero, so it passes the Alternating Series Test and converges. But if you take absolute values, you get the harmonic series ∑ 1/n, which diverges. Since the series itself converges but its absolute value series diverges, the convergence is conditional, not absolute. (If you’re curious, the sum equals -ln 2.)