Adding forever, ending somewhere
Fold a paper in half, then add half of the leftover, then half of that — the pieces sum toward one whole sheet even though you add forever. A convergent series is an endless sum that still lands on a finite total.
Adding infinitely many terms
A series is the sum of the terms of a sequence. It converges if its sequence of partial sums approaches a finite limit. A necessary condition is that the terms tend to zero — but that alone is not sufficient, as the harmonic series ∑1/n shows by diverging even though 1/n → 0.
The geometric series
∑ rⁿ converges precisely when |r| < 1, to the value 1/(1 − r). This is one of the few series with a clean closed form and a sharp convergence condition.
∑ (1/2)ⁿ from n = 0 = 1/(1 − 1/2) = 2. Halving repeatedly and summing approaches, but never exceeds, 2.
Lab · Converge or diverge?
- Consider the geometric series with ratio r = 1/3.
- Check whether |r| < 1.
- State whether it converges and, if so, to what.
What you should see: |1/3| < 1, so it converges; starting at n = 0 its sum is 1/(1 − 1/3) = 3/2.