Owl: A series is what happens when you keep adding terms forever: Σ aₙ. But "add forever" is not a casual instruction — it is a claim about the sequence of partial sums sₙ = a₁ + … + aₙ.
If those running totals approach a finite limit S, the series converges to S. If the totals wander or explode, the series diverges.
Picture a rain gauge that clicks once for each term. Sometimes the needle climbs toward a clear height. Sometimes it keeps climbing no matter how long you wait.
We'll build partial-sum tables, peek at telescoping cancellation, and meet the necessary test: if aₙ doesn't go to zero, the series cannot converge — and the converse trap that fools almost everyone.