Geometric Series - Calculus 2

Another special type of series you may encounter is what is known as a geometric series. A geometric series is a series where each subsequent term is found by multiplying the previous term by the same value known as the "common ratio".

A basic example of a geometric series would be 2+4+8+16+32+··· because each term after the first term is found by multiplying the previous term by 2. For example, 2 times 2 is 4, 2 times 4 is 8, 2 times 8 is 16 and so on. In this example, the common ratio would be 2 since that is the value that each term is multiplied by to get the next term.

In general, the form of a geometric series is fairly simple to identity, and once we know that a series is geometric, we can easily determine its convergence. The convergence is based on the value of the common ratio. If the absolute value of the common ratio is greater than 1, then the series will diverge, but if it is less than 1, then the series will converge to a specific value (we can calculate its sum!). We look at this in greater detail in this lesson, as well as look at a useful application of geometric series towards the end. So let's get to it!

In this lesson, you will learn:

• What a geometric series is and how to identify one
• How to determine the convergence of a geometric series and its sum (when convergent)
• How to use a geometric series to rewrite a repeating decimal as a fraction

I hope you find this video to be helpful!

-Josh