Summing a Geometric Series

A geometric series adds terms that each come from multiplying the last one by a fixed ratio r, like 5, 10, 20, 40. To sum the first n terms, call the sum S. Multiply S by r: every term shifts up one slot and lines up almost perfectly with the original sum. Subtract the two versions and nearly everything cancels, leaving only a first piece and a last piece. Solve for S and you get S = a(rⁿ - 1)/(r - 1), where a is the first term. That subtraction trick is the whole mechanism: it turns a long addition into two terms and a division.

The infinite version asks what happens as n grows without bound. If r is between -1 and 1, each new term shrinks toward zero, so the running total creeps toward a ceiling instead of growing forever. Plugging that shrinking behavior into the finite formula, the rⁿ piece vanishes, leaving S = a/(1 - r). That's why a stack of halves, 1 + 1/2 + 1/4 + 1/8 + ..., approaches 2 instead of blowing up: each added sliver is smaller than the gap still left to close.

This connects directly to arithmetic series, which sum by a different trick, pairing first and last terms, because a constant difference behaves differently than a constant ratio. It also sets up the convergence test: before summing an infinite geometric series, always check whether the ratio's absolute value is actually less than 1, since that single condition decides whether the sum even exists.