What is the radius of convergence for ∑ (-1)^n x^n /(n+1)?

• A. Radius 0
• B. Radius 1
• C. Radius ∞
• D. Radius 2

Answer: (B)

The radius of convergence is determined by how fast the coefficients a_n decay in the ratio test for a power series ∑ a_n x^n. Here a_n = (-1)^n/(n+1). Look at the ratio of consecutive coefficients in absolute value:

|a_{n+1}/a_n| = [1/(n+2)] / [1/(n+1)] = (n+1)/(n+2) → 1 as n → ∞.

Since the ratio test for the series ∑ a_n x^n gives the limit of |x| times this ratio, the limit is |x|. Convergence occurs for |x| < 1 and divergence for |x| > 1, so the radius of convergence is 1.

At the endpoints, x = 1 gives ∑ (-1)^n/(n+1), which converges by the alternating series test, while x = -1 gives ∑ 1/(n+1), which diverges. The radius remains 1, with the interval of convergence [-1, 1).