Can Adding a Sequence Forever Give a Finite Value?
Adding up the terms of a sequence, one after another, is called a series. Even though you're adding infinitely many terms, some series settle toward a fixed value (converge), while others grow without bound (diverge). Explore the geometric series sum formula, and how partial sums reveal whether a series converges or diverges.
A geometric series adds up every term of a geometric sequence with first term a and common ratio r (a + ar + ar² + ⋯). If |r| < 1, the terms get closer and closer to 0, and the sum converges to a value that, remarkably, can be found with the simple formula a ÷ (1−r). If |r| ≥ 1, the terms never go to 0, so the series diverges.