Infinite series and convergence

Pre-calculus showed that 1 + ½ + ¼ + … adds up to a finite number. Calculus makes that idea precise and asks the deciding question of Calc II: when does adding infinitely many terms give a finite total?

Partial sums and their limit

You can't add infinitely many numbers directly. Instead, add the first n and watch the running total — the partial sum Sₙ — as n grows:

• If the partial sums approach a finite limit L, the series converges, and we say its sum is L.
• If they grow without bound (or never settle), the series diverges.

A series is really a sequence of partial sums, and its sum is the limit of that sequence — the limit idea again, one level up.

The geometric series

The cleanest case, and the one worth memorizing. For a + ar + ar² + … :

So 1 + ½ + ¼ + … has a = 1, r = ½, and sum 1/(1 − ½) = 2. Each term is a shrinking fraction of the last, so the total is reined in.

A warning: small terms aren't enough

For a series to converge its terms must shrink to zero — but that alone isn't enough. The harmonic series 1 + ½ + ⅓ + ¼ + … has terms going to zero, yet its partial sums crawl to infinity. Deciding convergence is subtle, which is why Calc II builds a toolkit of convergence tests.

Why this matters

Infinite series are how we represent numbers and functions that have no finite formula: π, e, and the values your calculator returns for sin and ln are all sums of series. The next lesson uses them to rebuild whole functions — the payoff of the entire course.