Geometric Series Formula

Real Analysis · Axiom Academy

LESSON The Geometric Series Formula Why the infinite sum when |r| < 1 — and why it falls apart otherwise. 1. A Trick That Collapses the Sum Start with the partial sum of the first n terms — a finite sum we can handle. Write it out, then multiply the whole thing by r and line the two rows up: the partial sum (first n terms) Subtracting the second row from the first, every interior term has a twin and cancels. Only the very first term of S_n and the very last term of rS_n survive: The closed form has exactly one moving part: the term r^n . When |r| < 1 , raising r to higher powers drives it toward zero. Watch the partial sum S_n close the gap to its limit as that r^n piece evaporates: So the "gap" between S_n and the limit shrinks to nothing — the partial sums converge to a single finite number. The whole derivation of the infinite sum rests on . Remove that and the limit no longer exists. When the terms never shrink, so each one keeps adding at least as much as the last — and the partial sums run away: Terms shrink geometrically, , and the sum is the finite . Terms don't go to 0 , so S_n grows without bound (or oscillates) — no limit. For example has r = 2 : each term doubles, and the partial sums blow up. The formula simply does not apply. You derived the geometric series formula from its partial sums and saw exactly why convergence hinges on |r| < 1 . Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Real Analysis course.