Definition: A geometric series is the sum of terms in which each term is the previous one multiplied by a constant ratio r; it converges when the size of r is less than 1.
What is a geometric series?
Start with a first term a and multiply by a fixed ratio r each time: a, ar, ar2, and so on. The sum of the first n terms is
Sn = a(1 – rn)/(1 – r) for r not equal to 1.
What if it goes on forever?
If |r| < 1, the infinite sum exists and equals a/(1 – r). If |r| is 1 or more, it does not settle on a number.
Worked example: 40 + 10 + 2.5 + … has a = 40 and r = 1/4. The infinite sum is 40/(1 – 0.25) = 40/0.75 = 53.33…, or 160/3.
Finite check: the first 3 terms give 40(1 – 1/64)/(0.75), which is 52.5.
What do students mix up?
They apply the infinite formula when |r| is bigger than 1, such as r = 3. Mathbot’s rule: test the ratio before you use a/(1 – r).