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
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
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