Problem 113 - Precalculus with Limits

Question

In Exercises 113-116, find the indicated partial sum of the series. \( \displaystyle \sum_{i=1}^{\infty} 5 \left(\dfrac{1}{2} \right)^i \) Fourth partial sum

Step-by-Step Solution

Answer

The fourth partial sum of the series is 9.375.

Step 1: Identify 'a' and 'r'

The first term 'a' and the common ratio 'r' are critical to using the formula for the partial sum of a geometric series. For this series, 'a' equals 5 and 'r' equals 0.5.

Step 2: Insert values into the formula

Next, insert 'a' and 'r' into the formula for 'S_n' along with the value for 'n' which is 4 in this case (since we're finding the fourth partial sum): \( S_4 = 5 \dfrac{1 - (0.5)^4}{1 - 0.5} \)

Step 3: Calculate the partial sum

Use algebra to simplify the equation and calculate the value for 'S_4': \( S_4 = 5 [\dfrac{1 - 0.0625}{0.5}]= 5 \times 1.875 = 9.375 \)

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. 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Convergence of Series

Understanding when a series converges is critical in mathematical analysis. A series converges if the sum of its terms approaches a specific value as more terms are added, otherwise it diverges. For a geometric series to converge, the common ratio ewline ewlineewline ewlineewline ewlineewline ewlineewline gleewline gleewline gleewline gleewline gleewline gewline �ingewline � (ewline gewline gewline gewline gewline gewline �ingewline �)ewline gewline g�ing�ing�ing�ing�ing�ing� must be between -1 and 1. In our exercise, the common ratio ewline gewline gewline �ingewline �ingewline �ing�ing�ing�ing (gewline gewline gewline gewline �ingewline � 0.5gewline gewline gewline g�) satisfies this condition, implying that the infinite series has a finite sum. However, since we are calculating a partial sum—specifically, the fourth partial sum—we are concerned with a finite number of terms, and hence the series will definitely have a finite sum.

Sum Formula

The sum formula for a geometric series is incredibly useful when wanting to calculate the sum of the first 'n' terms. It is given by ewline gleewline gleewline gewline gewline � (ewline gewline gewline � )ewline gewline gewline gewline g� and requires that we know the first term 'a' and the common ratio 'r.' To find the sum 'S_n' of the first n terms of a geometric series with a non-zero common ratio, we use the formula ewline gewline gewline �ingewline �ingewline �ing�ing�ing�ing (gewline � (gewline gewline � )gewline g�).ewline gewline gewline g� When 'n' is finite and the value of 'r' is not equal to 1, we can calculate the sum of the first n terms. For example, in our original problem, we were asked to compute the fourth partial sum of a geometric series. By identifying 'a' as 5 and 'r' as 0.5, we inserted these values into the sum formula and simplified to find that the fourth partial sum is 9.375. This explicit formula is a powerful tool, allowing us to calculate the sum without needing to manually add each term.