Infinite Geometric Series
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When each term is a fixed fraction r of the one before, with |r| < 1, the infinite tail adds up to a finite value: a₁/(1 − r). It resolves Zeno's paradox — walking half the remaining distance forever, 1/2 + 1/4 + 1/8 + …, totals exactly (1/2)/(1 − 1/2) = 1, the whole journey.
The same collapse turns repeating decimals into fractions: 0.7777… is 0.7 + 0.07 + 0.007 + …, a geometric series with a₁ = 0.7 and r = 0.1, so S = 0.7/0.9 = 7/9.
Infinite Geometric Series
Where
- = Sum of the series
- = First term
- = Common ratio
Missing one of these? Work it out first, then come back
- Sum of the series — Sum of the Roots of a Quadratic, Arithmetic Series Sum
- First term — Arithmetic Sequence nth Term, Arithmetic Series Sum
- Common ratio — Geometric Sequence nth Term, Arithmetic Sequence nth Term