What is the radius of convergence of the power series ∑_{n=1}^∞ n x^n?

• A. 0
• B. 1
• C. Infinity
• D. 2

Answer: (B)

The radius of convergence is determined by how fast the coefficients grow relative to the powers of x. For the series ∑ n x^n, look at the ratio of successive terms t_n = n x^n. The ratio t_{n+1}/t_n equals (n+1)x^{n+1} / (n x^n) = (1 + 1/n) x, whose absolute value tends to |x| as n grows. The ratio test tells us the series converges when this limit is less than 1, so |x| < 1. That gives a radius of convergence R = 1.

Checking the endpoints clarifies the interval: at x = 1, the series becomes ∑ n, which diverges; at x = -1, it becomes ∑ n(-1)^n, whose terms do not go to zero, so it diverges as well. Thus the radius is 1, and the interval of convergence is (-1, 1).