Problem 58 - A Graphical Approach to College Algebra

Question

Find each sum that converges. $$\sum_{i=1}^{\infty} 5\left(-\frac{1}{4}\right)^{i-1}$$

Step-by-Step Solution

Answer

The sum of the series is 4.

Step 1: Identify the Type of Series

This is a geometric series of the form \( a r^{n-1} \), where \( a = 5 \) and \( r = -\frac{1}{4} \). A geometric series converges if the common ratio \( r \) is such that \( |r| < 1 \).

Step 2: Check for Convergence

The common ratio \( r = -\frac{1}{4} \) satisfies \( |-\frac{1}{4}| = \frac{1}{4} < 1 \). Therefore, the series converges.

Step 3: Use the Formula for Sum of an Infinite Geometric Series

The sum \( S \) of an infinite geometric series is given by the formula:\[S = \frac{a}{1 - r}\] where \( a \) is the first term and \( r \) is the common ratio.

Step 4: Calculate the Sum

Substitute \( a = 5 \) and \( r = -\frac{1}{4} \) into the formula to find the sum:\[S = \frac{5}{1 - (-\frac{1}{4})} = \frac{5}{1 + \frac{1}{4}}\]Simplify the denominator:\[\frac{5}{\frac{5}{4}} = 5 \times \frac{4}{5} = 4\]

Key Concepts

Convergence

In mathematics, convergence is a crucial concept when working with series, including geometric ones. The convergence of a series means that as we add more terms, the total value approaches a specific number rather than continuing to grow indefinitely. For a geometric series, we specifically look at the common ratio, denoted as \(r\), to determine if the series converges.
A geometric series converges when the absolute value of its common ratio is less than one. This can be expressed as \(|r| < 1\). In the given exercise, the common ratio \(r = -\frac{1}{4}\) meets this condition because \(|-\frac{1}{4}| = \frac{1}{4} < 1\). Thus, the series is convergent. This convergence suggests that, mathematically, we can sum an infinite number of terms to get a finite result. It's a fascinating aspect because it implies predictability and structure, even in endless quantities.

Infinite Series

An infinite series is when we sum an endless sequence of terms. For a geometric series like the one presented, this means calculating the total of a pattern that continues indefinitely. Unlike finite series, which have a set number of terms, infinite series proceed without limit. However, infinite series can still produce meaningful results through convergence.
In science and mathematics, infinite series are used across various disciplines. Whether modeling financial growth scenarios or describing physical phenomena, they are instrumental in illustrating concepts that stretch beyond simple sums. The geometric series in our exercise serves as a perfect example, successfully capturing an infinite summation through convergence. Each term in this type of series is derived by multiplying the previous term by a consistent value, the common ratio. As long as this common ratio's absolute value remains less than one, we can achieve an overall finite sum.

Sum of a Series

The sum of a series is the result of adding all its terms together. For geometric series, especially infinite ones that converge, we use a specific formula to find this sum. The formula for the sum of an infinite geometric series is given by:

• \(S = \frac{a}{1 - r}\)

Here, \(S\) represents the sum, \(a\) is the first term, and \(r\) is the common ratio.
Applying this to our exercise, where \(a = 5\) and \(r = -\frac{1}{4}\), we substitute these values into the formula:

• \[S = \frac{5}{1 - (-\frac{1}{4})} = \frac{5}{1 + \frac{1}{4}}\]

Simplifying the equation leads to:

• \[\frac{5}{\frac{5}{4}} = 5 \times \frac{4}{5} = 4\]

Thus, the sum of the series is 4. This calculation underscores the power of formulas to provide clear, finite answers even when dealing with infinite series. Understanding this method allows one to navigate explorations of geometric growth and decay in various contexts effectively.