Series Fundamentals

Calculus 2 · Axiom Academy

A series is just a sequence added up — and whether it has a sum is decided entirely by its sequence of partial sums. 1. From a Sequence to a Series A sequence is a list of numbers A series takes that list and adds the terms together. We write the sum compactly with sigma notation. The animation takes the separate terms and stacks them into one accumulating total — that stacking IS the series. The series — those terms summed 2. Partial Sums: Building Up the Total We can't add infinitely many terms at once, so we add them one at a time and record each running total. The n -th partial sum S_n stops after n terms. Watch each term a_k = 1/2^k lay down its own length on the line — the marker that lands after each one is S_n . Worked example — the terms a_n = 1/2^n 3. Convergence: Does the Series Have a Sum? Here is the central definition. A series isn't judged on its own — it is judged on its sequence of partial sums . If those partial sums settle onto a single number L , the series converges and we call L its sum. If they grow without bound or never settle, it diverges . Each plotted dot below is one S_n . 4. A Geometric Picture: Accumulated Area We can SEE a series. Draw each term a_n as a rectangle of width 1 and height a_n . As the sweep adds one rectangle at a time, the total shaded area is exactly the partial sum S_n . If that area approaches a limit, the series converges to it — and this same picture is what later powers the integral test.

This is the written version of the interactive lesson above. See the full Calculus 2 course.