1 · Observe

Blue dots: the terms a_n. Gold dots: the partial sums S_n.

2 · Manipulate

3 · Predict

Set geometric with a = 1, r = 0.5. Before looking: what is 1 + 1/2 + 1/4 + 1/8 + … forever? Now set r = 2. What happens to the partial sums?

4 · See

5 · Explain

An arithmetic series grows like n² (its partial sums form a parabola). A geometric series with |r| < 1 settles down: each new term is smaller than the last, so the total approaches S_∞ = a/(1 − r). If |r| ≥ 1 the terms never shrink and the sum diverges — there is no finite total.