Infinite series

"Adding infinitely many numbers" is not an operation anyone can perform, so the definition sidesteps it. Add the first n terms — a partial sum, perfectly ordinary — and then ask what those partial sums approach as n grows. If they approach a limit, the series converges and that limit is called its sum. If they do not, the series diverges. Everything about series is really about that sequence of partial sums.

Framed this way, the first surprise dissolves. 1/2 + 1/4 + 1/8 + … has partial sums 1/2, 3/4, 7/8, … creeping towards 1 without ever passing it. Infinitely many positive terms, finite total. The reason is that the terms shrink fast enough that what remains to add always fits in the gap left.

The second surprise is sharper. Terms must shrink to zero for convergence — but that is not enough. The harmonic series 1 + 1/2 + 1/3 + … has terms going to zero and diverges anyway: group them as 1/3 + 1/4 > 1/2, then 1/5 + … + 1/8 > 1/2, and so on forever, so the sum grows without bound however slowly. Deciding convergence therefore needs real tests — comparison, ratio, integral — rather than a glance at the terms. And convergent series are not all alike: conditionally convergent ones like 1 − 1/2 + 1/3 − … can be rearranged to sum to any value you choose, so even the commutativity of addition stops being safe once infinitely many terms are involved.