Calculus > Sum of a Convergent Geometric Series

In general, computing the sums of series in calculus is extremely difficult and is beyond the scope of a calculus II course. However, the geometric series is an exception. A geometric series can either be finite or infinite. A finite series converges on a number. For example, 1/2 + 1/4 + 1/8… converges on 1. An infinite geometric series does not converge on a number. For example, 10 + 20 + 20… does not converge (it just keeps on getting bigger).

The sum of a convergent geometric series can be calculated with the formula ^{a}⁄_{1-r}, where “a” is the first term in the series and “r” is the number getting raised to a power. A geometric series converges if the r-value is between -1 and 1.

## Sum of a Convergent Geometric Series: Example

**Sample problem:** Find the sum of the following geometric series:

Step 1: **Identify the r-value** (the number getting raised to the power). In this sample problem, the r-value is ^{1}⁄_{5}.

Step 2: **Confirm that the series actually converges**. The r-value for this particular series ( ^{1}⁄_{5}) is between -1 and 1 so the series does converge.

Step 3: **Find the first term**. Get the first term is obtained by plugging the bottom “n” value from the summation. The bottom n-value is 0, so the first term in the series will be (^{1}⁄_{5})^{0}.

Step 4: **Set up the formula** to calculate the sum of the geometric series, ^{a}⁄_{1-r}. “a” is the first term you calculated in Step 3 and “r” is the r-value from Step 1:

The sum of this particular geometric series is ^{5}⁄_{4}

*That’s it!*

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