infinite geometric series
An infinite geometric series adds endlessly many terms of a geometric sequence: a + ar + ar^2 + ar^3 + ... with no last term. Surprisingly, when each term is small enough relative to the last, this endless sum can still close in on a single finite number.
It converges exactly when the common ratio satisfies |r| < 1; then the sum is S = a_1 / (1 − r). The reason is that the term r^n in the finite-sum formula shrinks to 0 as n grows, so S_n = a_1·(1 − r^n)/(1 − r) heads toward a_1 / (1 − r).
If |r| ≥ 1 the terms do not shrink to 0, the partial sums never settle, and the series diverges — it has no finite sum, so the formula must not be used. The classic picture is 1/2 + 1/4 + 1/8 + ... , where the running totals creep ever closer to 1.
1/2 + 1/4 + 1/8 + ... has a_1 = 1/2 and r = 1/2, so S = (1/2)/(1 − 1/2) = 1.
With |r| < 1, the endless sum converges to a finite value.
A non-zero series can still have a finite sum — convergence is about the partial sums settling down, not about the terms ever literally reaching zero.