A geometric series is a sum of numbers such that the ratio between consecutive terms is constant. For instance, 1/2 + 1/4 + 1/8 + … is a geometric series. The harmonic series is the sum of all numbers 1/n for nn = 1, 2, 3, …. In this activity, you can see a representation of three geometric series and the harmonic series.
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Choose the type of series to view: a geometric series with ratio 1/2, 1/3, or 1/4; or, the harmonic series.
Repeatedly click Show Next Step to see the series build, term by term. The number of terms will be indicated in the "n =" box, and the sum of the terms will be shown in the "S =" box.
Use Zoom In/Out to increase or decrease the scale, and use Reset to start over. The "Enable Animations" checkbox allows you to turn off the animations.
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Choose "Geometric Series ‑ 1/2," which will build a geometric series for which the ratio between terms is 1/2. That is, it will generate the sum
1/2 + 1/4 + 1/8 + 1/16 + ….
As more terms are added, how does the sum change? Do you think the sum will continue to increase forever, or is there some limit to how large the sum will get?
Do the same with "Geometric Series ‑ 1/3" and "Geometric Series ‑ 1/4."
- Do the sums of these series have a limiting value, or do they continue forever?
- What pattern do you notice in the sum for all three series?
Finally, build the harmonic series. How is this series different from the others?
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