Infinite Geometric Series

This formula gives the total of a geometric pattern that keeps shrinking by the same factor each time, so the added terms become too small to change the sum. The reason it works is that if you multiply the whole infinite sum by the common ratio r and subtract, almost all terms cancel, leaving only the first term a; this cancellation is valid only when the terms actually approach 0, which happens when |r|<1. In that case, the infinite sum settles to a finite limit of a/(1-r).

Use this whenever a quantity repeats with constant proportional decay and you need the long-run total, such as discounted cash flows, repeated reflections, exponential-decay approximations, or accumulating probabilities in a geometric process. It is especially useful when the pattern is geometric and extends indefinitely but still converges.

• Using the formula when |r| is 1 or greater, which makes the series diverge instead of converging
• Forgetting that a is the first term of the series, not the ratio between terms
• Substituting r with the second term instead of the common ratio between consecutive terms