Evaluating Σ(2/3)ⁿ from n=0 to ∞

Calculus 2 · Axiom Academy

Identify, test for convergence, and sum an infinite geometric series step-by-step. Evaluate the infinite series . Decide whether it converges, and if it does, find its exact sum. Nice work! You evaluated an infinite geometric series from start to finish. Here's what we used: Spot the pattern: a geometric series has a constant ratio between consecutive terms — here each term is the previous one times . Check convergence first: a geometric series converges only when . Always verify this before reaching for the formula. Apply the formula: for a convergent geometric series starting at n=0 , the sum is , where a is the first term and r is the common ratio. Watch it converge: the partial sums creep steadily toward 3 — exactly the value we computed. Mind the starting index: because the series starts at n=0 , the first term is . If it started at n=1 , the first term would be instead. This same technique shows up everywhere in Calculus 2: compound interest, probability, and physical processes with diminishing effects all lean on convergent geometric series.

This is the written version of the interactive lesson above. See the full Calculus 2 course.