What is the radius of convergence of the Maclaurin series for ln(1+x)?

• A. 1
• B. 0
• C. ∞
• D. 2

Answer: (A)

The radius of convergence is controlled by how far you can move from 0 in the complex plane before the function stops being analytic. ln(1+x) has a singularity at x = -1, so the nearest singularity to 0 is at distance 1. That makes the Maclaurin series converge for |x| < 1. The actual series is ln(1+x) = x − x^2/2 + x^3/3 − x^4/4 + ..., valid for |x| < 1, and it happens to converge at x = 1 (to ln 2) but diverges at x = −1. Therefore the radius of convergence is 1.