Problem 106 - Precalculus with Limits

Question

In Exercises 93 - 106, find the sum of the infinite geometric series. \( -\dfrac{125}{36} + \dfrac{25}{6} - 5 + 6 - \cdots \)

Step-by-Step Solution

Answer

The sum of the infinite geometric series is -125/6.

Step 1: Identify the first term (a)

The first term \( a \) of the series is -125/36. It is the first number in the series.

Step 2: Find the common ratio (r)

The common ratio \( r \) is found by dividing the second term by the first term. So, \( r = -\dfrac{25}{6} / -\dfrac{125}{36} = 5/6 \)

Step 3: Calculate the sum of the series

Use the sum formula of an infinite series \( S = \dfrac{a}{1 - r} \). Substituting the given values: \( S = \dfrac{-125/36}{1 - 5/6} = -\dfrac{125}{36} \cdot \dfrac{6}{1} = -\dfrac{125}{6} \)

Key Concepts

Understanding the Common Ratio

In an infinite geometric series, each term after the first is determined by multiplying the preceding term by a constant, known as the common ratio. Identifying this ratio is a critical step in analyzing or summing the series.
To find it, you simply divide any term in the series by the term that comes before it. When dealing with the series from the exercise, we used the second term \(-\frac{25}{6}\) and divided it by the first term \(-\frac{125}{36}\).
This gives us the common ratio, \(r = \frac{5}{6}\), which is crucial for determining subsequent terms and understanding the behavior of the series.

• A common ratio \(|r| < 1\) implies the series terms are decreasing.
• If \(|r| > 1\), the terms are increasing, meaning the series does not converge.
• For our series, \(r = \frac{5}{6}\) indicates the terms will decrease in size.

This ratio serves as the foundation for calculating the series' sum and understanding its convergence properties.

Exploring the Sum Formula for Infinite Series

Once the common ratio is found and it's confirmed that \(|r| < 1\), we can calculate the sum of the infinite geometric series using the sum formula:
The formula is \(S = \frac{a}{1 - r}\), where \(a\) is the first term of the series. In the example exercise, the first term \(a\) is \(-\frac{125}{36}\), and we calculated \(r\) to be \(\frac{5}{6}\).
Plugging these values into the sum formula gives us: \[ S = \frac{-125/36}{1 - 5/6} = \frac{-125/36}{1/6} = -\frac{125}{6} \]
This formula is remarkable because it enables you to find the sum of an infinite number of terms, assuming the series converges. It becomes particularly powerful when the common ratio remains less than one in absolute value, maintaining the decreasing nature of the terms.

Geometric Series Convergence Explained

A geometric series converges when the terms eventually approach zero. The most important factor for convergence in an infinite geometric series is the absolute value of the common ratio \(|r|\).
The series will converge if \(|r| < 1\). This means that each additional term will become smaller and smaller, bringing the sum closer to a finite number.
For the provided series with \(r = \frac{5}{6}\), it's clear that \(|r|\) is indeed less than 1, so we can safely say that the series converges.
Understanding convergence is pivotal because it assures us that the sum we calculate using the sum formula will be meaningful and accurate. Here are some key takeaways when evaluating convergence:

• Check \(|r| < 1\) to ensure convergence.
• A convergent series lets us apply the sum formula confidently.
• With convergence confirmed, the series offers a single summable limit.

This understanding equips you with the tools to tackle infinite series confidently, ensuring you know when solutions are reliable and applicable.