Geometric series
A series where each term is a fixed multiple r of the one before. It adds to a / (1 - r) when |r| < 1, and diverges otherwise.
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Analogy
A bouncing ball that rises to half its previous height each time: the total bounce height is finite.
Example
First term a = 3. Ratio r = 0.5.Sum = 3 / (1 - 0.5).= 3 / 0.5.Sum = 6.
In practice
The most important series, used for loans, savings, decimals like 0.333..., and comparisons.
Practice
Flashcards
- Sum of a geometric series with first term a and ratio r?a/(1 - r), when |r| < 1.
- What is 3 + 1.5 + 0.75 + ... equal to?6.
Interview questions
- Find the sum of 3 + 1.5 + 0.75 + ... and explain why it is valid.
- Geometric with a = 3, r = 0.5
- |r| < 1 so the series converges
- Sum = a/(1 - r) = 3/0.5 = 6
- Partial sums approach 6
Questions people ask
How do I find the ratio of a geometric series?
Divide any term by the one before it. In 3, 1.5, 0.75, the ratio is 1.5 over 3, which is 0.5. Check with another pair, 0.75 over 1.5, to see it is the same. If the ratios differ, the series is not geometric.
What happens if the ratio is 1 or bigger?
The terms do not shrink, so the total cannot settle. With r = 1 every term is the same size, so the sum grows forever. With r = 2 the terms double each time. The formula a over 1 minus r only works when absolute r is less than 1.