The ancient Greek philosopher Zeno posed this paradox: "Swift Achilles can never catch a tortoise that starts a little ahead of him." By the time Achilles reaches the spot where the tortoise was, the tortoise has already moved a bit further ahead; by the time he reaches that new spot, the tortoise has moved a little further still… and since this keeps happening forever, the argument goes, he can never catch up. Of course, in reality Achilles overtakes the tortoise in no time at all. The trick hiding inside this paradox is a series — the idea of "adding infinitely many numbers together."
A series is what you get when you add up the terms of a sequence one after another. Given a sequence a₁, a₂, a₃, …, the sum a₁+a₂+a₃+⋯ is called a series. At first it seems like adding infinitely many numbers must always give an infinitely large answer, but surprisingly, some series settle on a finite value. The classic example is a geometric series. Add up every term of a geometric sequence with first term a and common ratio r — a+ar+ar²+ar³+⋯ — and as long as |r| < 1, this infinite sum surprisingly converges to the exact value a ÷ (1−r).
Take a square and shade half of it (1/2), then shade half of what's left (1/4), then half of that (1/8)… keep this up forever, and the sum of the shaded areas, 1/2+1/4+1/8+⋯, never goes over 1 but gets endlessly closer to it. This is in fact a geometric series with a=1/2 and r=1/2, so the formula gives (1/2)÷(1−1/2) = 1. You add infinitely many terms, and the answer lands exactly on 1. Zeno's paradox works the same way: even if you split the time it takes Achilles to catch the tortoise into infinitely many tiny pieces and add them up, that sum converges to a finite time — the sum of a geometric series. That's why Achilles really can catch the tortoise in a finite amount of time.
So how do you tell whether a given series converges or diverges? The key is the partial sum. For the series a₁+a₂+⋯, define the partial sum up to the nth term as Sₙ = a₁+a₂+⋯+aₙ; these Sₙ values themselves form a sequence. A series converging means precisely that "this sequence of partial sums {Sₙ} converges to some value as n→∞." In other words, the convergence or divergence of a series turns directly into a question about the limit of a sequence. For example, a series like 1/(1×2) + 1/(2×3) + 1/(3×4) + ⋯ can be rewritten term by term as 1/n − 1/(n+1); the middle terms then cancel out one after another in what's called a "telescoping series," leaving the partial sum as 1 − 1/(n+1), which converges to 1.
There are also examples that diverge. Σ 1 (a series where every term is 1) has partial sum Sₙ=n, which grows without bound, so of course it diverges. Here's one very important fact: for a series to converge, its terms aₙ must converge to 0. If the terms don't go to 0, the series is guaranteed to diverge (though the reverse isn't true — terms going to 0 doesn't always mean the series converges. That's an interesting exception you'll run into again when you study series more deeply).
On our activity page, you can change the first term and common ratio with sliders to check that the sum of a geometric series really does match the formula, and watch the partial sums of several series build up on a bar graph to compare convergence and divergence for yourself.