Infinite Geometric Series

Algebra 2 · Axiom Academy

LESSON Infinite Geometric Series Can a sum of infinitely many terms be finite? Meet the convergence condition and the formula S = a_1/(1-r) . 1. Can an Infinite Sum Be Finite? Compare two series. Halving each time, , the running totals creep up toward 1 and never pass it. Doubling each time, , the running totals explode. Whether the partial sums S_n settle at a point or run away is the whole question. Shrinking terms settle; growing terms don't Every finite geometric sum has a clean closed form. Summing the first n terms gives . The only place n appears is in the tail . Send n to infinity: if , then , and the whole expression settles onto a single number. Taking that limit turns the partial-sum formula into the sum of the whole infinite series — valid only while . Sum of an infinite geometric series a_1 = 8 , . Since , the series converges: , — the halving series from Step 1 — sums to exactly 1: 4. When the Formula Fails: Divergence If the terms do not shrink, there is no finite total — and becomes a trap : it still returns a number, but a meaningless one. No shrinking tail, no finite sum Each term is larger than the last; the partial sums increase without bound. Example r = 2 : . Every term equals a_1 , so . The sum grows past any bound. Terms flip sign; S_n bounces and never settles on a limit. For r = 2 the formula returns -1 — impossible, since is a sum of growing positive terms. The formula is valid only when .

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