Problem 63 - A Graphical Approach to College Algebra

Question

Find each sum that converges. $$\sum_{i=1}^{\infty}\left(\frac{1}{5}\right)\left(-\frac{1}{2}\right)^{i-1}$$

Step-by-Step Solution

Answer

The series converges to \(\frac{2}{15}\).

Step 1: Identify the Type of Series

The series given is \(\sum_{i=1}^{\infty} a \cdot r^{i-1}\), where \(a = \frac{1}{5}\) and \(r = -\frac{1}{2}\). This matches the form of a geometric series.

Step 2: Recall the Geometric Series Convergence Criteria

A geometric series \(\sum a \cdot r^{i-1}\) converges if \(|r| < 1\). Our series has \(r = -\frac{1}{2}\), so \(|r| = \frac{1}{2} < 1\). Therefore, the series converges.

Step 3: Calculate the Sum of the Convergent Geometric Series

The sum \(S\) of an infinite geometric series \(\sum a r^{i-1}\) is given by \(S = \frac{a}{1-r}\), provided \(|r| < 1\). Here, \(a = \frac{1}{5}\) and \(r = -\frac{1}{2}\).

Step 4: Substitute Values into the Sum Formula

Substitute \(a = \frac{1}{5}\) and \(r = -\frac{1}{2}\) into the formula to find the sum: \[ S = \frac{\frac{1}{5}}{1 - (-\frac{1}{2})} = \frac{\frac{1}{5}}{1 + \frac{1}{2}} = \frac{\frac{1}{5}}{\frac{3}{2}} \].

Step 5: Simplify the Expression

Simplify \(\frac{\frac{1}{5}}{\frac{3}{2}}\) by multiplying the numerator and the denominator by 2: \[ S = \frac{1}{5} \times \frac{2}{3} = \frac{2}{15} \].

Key Concepts

Convergence Criteria

When dealing with geometric series, it's crucial to determine if the series converges. Convergence tells us if the sum of the series leads to a finite number or not. For infinite geometric series, the series converges only if the absolute value of the common ratio \(r\) is less than 1. This is because when \(|r| < 1\), each successive term becomes smaller and smaller, eventually approaching zero.

This makes the entire series stabilize to a fixed sum. If \(|r| \geq 1\), the series doesn't settle to a finite sum and diverges instead, heading towards infinity or oscillating. In our example, the series given has \(r = -\frac{1}{2}\), and clearly \(| -\frac{1}{2} | = \frac{1}{2} < 1\). Therefore, this series converges, allowing us to calculate its sum.

Infinite Series

An infinite series, as the name suggests, consists of an endless sequence of terms added together. Infinite series come in various forms, but the geometric series is a special, widely-studied type distinguished by its common ratio. Each term in the series is a fixed multiple of the previous one, determined by this ratio.

In our exercise, the series is infinite because it sums terms from \( i=1 \) to infinity with each term constructed using a pattern: \(a \cdot r^{i-1}\). Here, \(a\) is the first term, and \(r\) is the common ratio."

• For our series: \(a = \frac{1}{5}\), \(r = -\frac{1}{2}\)
• The terms progressively decrease due to \(r\) being a fraction less than one in magnitude

This structure forms the backbone of many real-world phenomena, such as economic models, signal processing, and more, where elements naturally repeat and degrade over time.

Sum Formula

Once we've established a geometric series converges, we can turn to the sum formula to find its total value. For an infinite geometric series, this sum is calculated using the formula:

\[ S = \frac{a}{1 - r} \]

where \(S\) is the sum, \(a\) is the first term, and \(r\) is the common ratio.

In the given problem, the first term is \(a = \frac{1}{5}\) and the common ratio is \(r = -\frac{1}{2}\). Substituting these values into the formula gives:

• First, adjust for the common ratio sign: \(1 - ( -\frac{1}{2}) = 1 + \frac{1}{2} = \frac{3}{2}\)
• Calculate: \(S = \frac{\frac{1}{5}}{\frac{3}{2}}\)
• Simplify to get: \(S = \frac{2}{15}\)

Using this formula highlights the beauty of geometric series—a seemingly infinite set can be condensed into a simple expression, making complex calculations manageable.