A geometric series is the sum of the terms in a geometric sequence. In a geometric sequence, each term is found by multiplying the previous term by a fixed number called the common ratio. When you add up the terms of such a sequence, you get a geometric series.
For example, the sequence 2, 4, 8, 16 is a geometric sequence with a common ratio of 2. If you add those terms together, 2 + 4 + 8 + 16, you get a geometric series.
Refresher: A geometric sequence is just a list of numbers with a common ratio. A geometric series is the sum of that list.
Geometric series can be either finite or infinite:
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Finite geometric series: Has a set number of terms. You can find the total using a specific formula to avoid adding each term one by one.
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Infinite geometric series: Continues forever. If the terms get smaller and smaller (meaning the absolute value of the common ratio is less than 1), the series approaches a fixed value, called its sum. If the terms do not shrink, the series does not have a finite sum and is said to diverge.
Use the geometric sum formula to calculate the total for both types of series.
Geometric series are used in many areas of math and science, including finance (like calculating interest or loan payments), physics, and computer science. They help model repeated multiplication or exponential growth and exponential decay.
If you add the infinite series 1 + 1/2 + 1/4 + 1/8 + 1/16 + …, each term is half the one before it. This geometric series has a common ratio of 1/2 and adds up to 2, even though the terms keep going forever. The shrinking values get closer and closer to 0, and the total gets closer and closer to 2.
In physics, geometric series appear when modeling how light reflects between two surfaces with partial reflectivity, such as in a beam splitter. Each reflection transmits a smaller portion of the original light. If 60% of the light passes through on each bounce, the total amount of light transmitted over time forms a geometric series: 0.6 + 0.24 + 0.096 + … This infinite geometric series has a common ratio of 0.4 and converges to a total transmission of 1. In other words, all the light is eventually transmitted—just in smaller and smaller pieces.
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