Problem 17 - Thomas Calculus

Question

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty}-\frac{1}{8^{n}} $$

Step-by-Step Solution

Answer

The series converges because it is a geometric series with a common ratio less than 1.

Step 1: Identify the Type of Series

The given series is \( \sum_{n=1}^{\infty} -\frac{1}{8^n} \). This series is an infinite geometric series because it has the form \( \sum_{n=1}^{\infty} ar^n \) where \( a = -\frac{1}{8} \) and \( r = \frac{1}{8} \).

Step 2: Determine the Common Ratio

Identify the common ratio \( r \) of the geometric series from the given terms. The common ratio \( r \) for this series is \( \frac{1}{8} \), calculated by dividing the second term by the first term or using the expression from the series' general form.

Step 3: Assess the Convergence Condition

A geometric series \( \sum_{n=1}^{\infty} ar^n \) converges if the absolute value of the common ratio \( |r| < 1 \). Here, \( |r| = \left| \frac{1}{8} \right| = \frac{1}{8} < 1 \), indicating that the series converges.

Step 4: State the Conclusion

Since the series is geometric with a common ratio \( \left| \frac{1}{8} \right| < 1 \), it converges. The convergence is determined by the value of \( r \) and the convergence condition for geometric series.

Key Concepts

Geometric Series

In mathematics, a geometric series is crucial for understanding sequences of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. These series often appear in various real-world situations, such as calculating interest, population growth, or the sum of a sequence of payments.

A geometric series can be represented as:

• First term: \( a \)
• Common ratio: \( r \)
• General term: \( ar^n \) where \( n \) is a non-negative integer

To express this in standard form, the series is \( a + ar + ar^2 + ar^3 + \ldots \), and when written as an infinite series, it is \( \sum_{n=0}^{\infty} ar^n \).
A key point in examining geometric series is determining convergence or divergence. This depends on the common ratio \( r \):

• If \(|r| < 1\), the series converges, meaning it approaches a specific value.
• If \(|r| \geq 1\), the series diverges, which means it doesn't settle to a limit.

Understanding the behavior of geometric series is essential in both theoretical aspects of mathematics and practical scenarios.

Infinite Series

An infinite series, as the name suggests, is a sum of infinitely many terms. To make sense of such series, we observe their behavior as the number of terms grows indefinitely. Calculating the sum of all these terms can reveal important properties, particularly about how they converge or diverge.

The series \( \sum_{n=1}^{\infty} -\frac{1}{8^n} \) represents an infinite series where terms continue to add up without bound. When dealing with infinite series, the focus is often on whether these sums approach a finite limit or not.
For an infinite series to converge:

• The sequence of partial sums \( S_n = a_1 + a_2 + \cdots + a_n \) must approach a specific finite number as \( n \) approaches infinity.
• For example, in geometric series like \( \sum_{n=0}^{\infty} ar^n \), convergence typically hinges on the common ratio being within a certain range (specifically, \(|r| < 1\)).

If an infinite series does not meet these conditions, it diverges, indicating that the sum increasingly grows without reaching a finite boundary. Understanding infinite series is crucial in calculus, as it forms the foundation of series expansions and numerous approximation methods.

Common Ratio

The common ratio is a vital part of understanding geometric series, serving as the factor by which each term of the series is multiplied to yield the next. It's a constant that defines the trajectory and behavior of the series. Calculating it correctly is necessary for evaluating whether a series will converge or diverge.

In our case, the series \( \sum_{n=1}^{\infty} -\frac{1}{8^n} \) has a common ratio identified as \( r = \frac{1}{8} \).
To find the common ratio:\

• Take any term in the series (except the first) and divide it by the preceding term.
• For example, calculate \(-\frac{1}{8^2} \div -\frac{1}{8^1} = \frac{1}{8}.\)

Understanding the absolute value of \( r \) is particularly important:

• If \(|r| < 1\), it indicates that the terms will progressively get smaller, causing the series to converge.
• Otherwise, if \(|r| \geq 1\), the terms could either stay the same or grow larger in magnitude, leading the series to diverge.

Knowing whether \( |r| \) meets the convergence criteria is key when analyzing geometric series in mathematical problems.