Series and Convergence

Walkthrough

Determine whether the geometric series Σ (from n=0 to ∞) of (2/3)ⁿ converges, and find its sum if it does.

1. 1.Recall the geometric series convergence test: a geometric series Σ arⁿ converges if and only if |r| < 1.
2. 2.Identify the common ratio: r = 2/3, so |r| = 2/3 < 1.
3. 3.Since |r| < 1, the series converges. The first term is a = (2/3)⁰ = 1.
4. 4.Apply the sum formula: S = a / (1 − r) = 1 / (1 − 2/3) = 1 / (1/3).
5. 5.Compute: 1 / (1/3) = 3. So the sum is 3.