When does an infinite geometric sequence have a finite sum?

The convergence condition for an infinite geometric sum

In a geometric sequence, every term is calculated by multiplying the previous term by a constant value called the common ratio, written as q. You find q by dividing any term by the one right before it, never by subtracting.

Such an infinite progression produces a finite total only when the absolute value of the common ratio is strictly less than one: |q| < 1 (or -1 < q < 1, with q != 0). When this condition holds, each term becomes smaller in magnitude than the last, shrinking toward zero rapidly enough that the total sum approaches a single finite limit. If |q| is greater than or equal to one, the terms stay the same size or grow larger, so the sum diverges and no finite sum exists.

The formula and a real-world bounce example

When the condition |q| < 1 is satisfied, the infinite sum S equals the first term a1 divided by (1 - q): S = a1 / (1 - q).

Consider a rubber ball dropped from a height of 3 meters that rebounds to two-thirds of its previous height after each bounce. The first rebound reaches 3 * (2/3) = 2 meters. The second reaches 2 * (2/3) = 4/3 meters, and so on. Here, the first term a1 is 2, and the common ratio q is 2/3. Because |2/3| < 1, the total sum of all rebound heights is finite: S = 2 / (1 - 2/3) = 2 / (1/3) = 6 meters. Even though the ball rebounds infinitely many times in theory, the total rebound distance cannot exceed 6 meters.

Common pitfalls to avoid

Watch out for three frequent mistakes. First, do not plug values into the infinite sum formula if |q| is greater than or equal to one. The formula will spit out a number, but that number is mathematically invalid because the series diverges.

Second, do not reject negative ratios. If q is -1/2, its absolute value is 1/2, which is less than one. The terms alternate between positive and negative values while shrinking, and the formula S = a1 / (1 - q) works properly.

Finally, when solving for q between non-consecutive terms, remember that taking an even root yields both positive and negative solutions. If a1 = 5 and a3 = 45, then q squared equals 9, giving q = 3 or q = -3.