Finding Sums of Infinite Series
When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first terms of a geometric series.
We will examine an infinite series with . What happens to as increases?
The value of decreases rapidly. What happens for greater values of
As gets very large, gets very small. We say that, as increases without bound, approaches 0. As approaches 0, approaches 1. When this happens, the numerator approaches . This give us a formula for the sum of an infinite geometric series.
A General Note: Formula for the Sum of an Infinite Geometric Series
The formula for the sum of an infinite geometric series with isHow To: Given an infinite geometric series, find its sum.
- Identify and .
- Confirm that .
- Substitute values for and into the formula, .
- Simplify to find .
Example 7: Finding the Sum of an Infinite Geometric Series
Find the sum, if it exists, for the following:Solution
- There is not a constant ratio; the series is not geometric.
- There is a constant ratio; the series is geometric. and , so the sum exists. Substitute and into the formula and simplify to find the sum:
- The formula is exponential, so the series is geometric with . Find by substituting into the given explicit formula:
Substitute and into the formula, and simplify to find the sum:
- The formula is exponential, so the series is geometric, but . The sum does not exist.
Example 8: Finding an Equivalent Fraction for a Repeating Decimal
Find an equivalent fraction for the repeating decimalSolution
We notice the repeating decimal . so we can rewrite the repeating decimal as a sum of terms..
Looking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to 0.1 in the second term, and the second term multiplied to 0.1 in the third term.
![0.3 repeating equals 0.3 + 0.1 + 0.3 (the first term) + 0.1. (0.1)(0.3) (the second term)](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/1535/2015/11/25213751/Screen-Shot-2015-09-15-at-10.22.06-PM.png)
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