Problem 11 - Entrance Test

Determine the interval of convergence for the power series ∑ (n=0 to ∞) (x - 2)^n / (n + 1).

A. [1, 3)B. (1, 3]C. (1, 3)D. [1, 3]E. (-∞, ∞)

Correct: A

Using the Ratio Test: lim (n→∞) |(x - 2)^(n+1) / (n + 2) * (n + 1) / (x - 2)^n| = lim (n→∞) |(x - 2) * (n + 1) / (n + 2)| = |x - 2|. For convergence, |x - 2| < 1, so -1 < x - 2 < 1, which gives 1 < x < 3. Check endpoints: x = 1: ∑ (-1)^n / (n + 1) converges by the Alternating Series Test. x = 3: ∑ 1 / (n + 1) diverges (harmonic series). Therefore, the interval of convergence is [1, 3).