← Unit 10 Guide

Series Foundations

Start here. Before the convergence tests and Taylor series, get the core idea down cold: what a series even is, and what it means for an infinite sum to "add up."

The whole unit rests on one surprising question: can you add up infinitely many numbers and get a finite total? Sometimes yes, sometimes no — and the way you tell the difference is by watching the partial sums: add the terms one at a time and see where the running total heads. This page lets you watch that happen.

Watch a series add up

Pick a series, then add its terms one at a time. The gold dots are the running total after each term — the partial sums. Watch whether they settle toward a line (converges) or run off and never settle (diverges).

each term aₙ partial sum Sₙ (running total) the limit

Your call first — does this series converge or diverge?

Choose a series

Tip: the harmonic series is the famous trap — its terms shrink to zero, yet the total still grows forever. "Terms → 0" is necessary but not enough for convergence.

The four ideas that everything else builds on

Sequence vs. series

A sequence is a list of numbers. A series is what you get when you add that list up. Same numbers — one is the list, the other is the sum.

Partial sums

You can't add infinitely many numbers at once, so you add them one at a time. The total after n terms is the n-th partial sum, Sₙ. The dots in the graph above are exactly these.

Convergence

If the partial sums home in on a single finite number as you add more terms, the series converges to that number — its sum.

Divergence

If the partial sums grow without bound, or bounce around and never settle, the series diverges — it has no finite sum.

Got the foundation? Next, learn to choose the right convergence test in the Convergence Compass, or jump to the Unit 10 Guide.