5. Geometric Series
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Let's look at an example of why this is true. Consider the following two infinite geometric series.
Series | Common Ratio | Series Type |
---|---|---|
1+2+4+8+16+… | r=2 | Divergent |
1+21+41+81+161+… | r=21 | 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 2.
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 ∣r∣>1. Now, let's think about the series that converges. This time, each new term is half the value of the previous term.
approaching the limitof the sum and will be true for any geometric series such that ∣r∣<1.