Pearson Algebra 2 Common Core, 2011
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Pearson Algebra 2 Common Core, 2011 View details
5. Geometric Series
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Exercise 6 Page 598

An infinite geometric series is the sum of the terms of an infinite geometric sequence.
The common ratio of a geometric series is The value of will determine whether the series has a sum.
If the absolute value of the common difference is less than the series converges and therefore has a sum.

Extra

An example

Let's look at an example of why this is true. Consider the following two infinite geometric series.

Series Common Ratio Series Type
Divergent
Convergent

Notice that each term in the series that diverges is twice the value of the previous term. In this example, each term grows by a factor of

This pattern will continue infinitely and the sum will also keep growing. The same will be true for any series that has a ratio such that Now, let's think about the series that converges. This time, each new term is half the value of the previous term.

This pattern will also continue infinitely. As this happens, the sum will increase by a smaller degree each time until, eventually, adding each new term results in a negligible change. Look at how small the values of the and terms in this series are.
This is called approaching the limit of the sum and will be true for any geometric series such that