Problem 19 - Thomas Calculus

Question

Which of the series in Exercises 13 46 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 is a geometric series of the form \( \sum_{n=1}^{\infty} ar^n \), where \( a = -\frac{1}{8} \) and \( r = \frac{1}{8} \).

Step 2: Check Geometric Series Convergence Criteria

A geometric series \( \sum_{n=1}^{\infty} ar^n \) converges if the absolute value of the common ratio \( |r| < 1 \). In this case, \( |r| = |\frac{1}{8}| = \frac{1}{8} \), which is less than 1, so the series converges.

Step 3: Determine the Sum of the Series

Since the series converges, we can find its sum using the formula for an infinite geometric series: \( S = \frac{a}{1-r} \). Here, \( a = -\frac{1}{8} \) and \( r = \frac{1}{8} \), so the sum is \( S = \frac{-\frac{1}{8}}{1-\frac{1}{8}} = \frac{-\frac{1}{8}}{\frac{7}{8}} = -\frac{1}{7} \).

Key Concepts

Series Convergence

Understanding whether a series converges or diverges is a foundational skill in calculus and helps us comprehend the sum behaviors of infinite solutions. When we talk about a series converging, we mean that its terms approach a single finite value as more terms are added. This is particularly useful in infinite series, where determining the sum is not as straightforward as adding a few numbers. Convergence ensures that we have a bound and predictable outcome.

• A series may converge if it approaches a limit as the number of terms progresses towards infinity.
• Divergence, on the other hand, implies the series grows without bound, jumping to infinite solutions that cannot be pinned down to a single finite value.
• It's essential to identify the nature of a series to determine whether its sum reaches a finite number or keeps expanding indefinitely.

Understanding the concept of series convergence gives us control in mathematical analysis, as it teaches us how to handle infinite, boundless additions smartly and effectively.

Geometric Series Formula

The geometric series is a special kind of series where each term is a constant multiple, termed as the common ratio, of the preceding term. This simple pattern makes it possible for us to analyze the series effectively.

• Geometric Series: A series of the form \( \sum_{n=0}^{\infty} ar^n \), where \( a \) is the initial term, and \( r \) is the common ratio.
• The formula for finding the sum of a finite geometric series is \[ S_n = \frac{a(1-r^n)}{1-r} \text{, if } r eq 1 \].

In our original exercise, we deal with an infinite geometric series characterized by \( a = -\frac{1}{8} \) and \( r = \frac{1}{8} \). The simple structure of a geometric series allows for easy manipulation, particularly when working towards determining convergence or calculating sums.

Infinite Series Sum

For infinite series, calculating a sum might seem impossible at first glance because they contain an unending number of terms. Fortunately, there's a way to handle them through mathematical tools and formulas.

• An infinite geometric series with a common ratio \(|r| < 1\) converges and has a finite sum.
• The formula for the sum of an infinite geometric series is \( S = \frac{a}{1-r} \).
• Applying this to our example \( \sum_{n=1}^{\infty} -\frac{1}{8^n} \), the sum becomes \( S = \frac{-\frac{1}{8}}{1-\frac{1}{8}} = -\frac{1}{7} \). This computation reveals that, despite having infinite terms, the series is directed towards a specific number.

The power of this formula is it provides a definitive sum for series where manual addition is non-existent, giving immediate clarity.

Absolute Value Criterion for Convergence

The idea behind using the absolute value criterion for convergence is a handy tool. This method focuses on the comparison of the magnitude of the terms rather than their direction (positive or negative).

• For a geometric series to converge, the absolute value of the common ratio, \( |r| \), must be less than 1.
• In the case of \( r = \frac{1}{8} \), we find \(|r| = |\frac{1}{8}| = \frac{1}{8}\), clearly less than 1, confirming convergence.
• Effectively, this criterion simplifies understanding convergence by eliminating the need to consider complex dynamics or large calculations for checking each term.

This approach is one of the most straightforward methods when dealing with geometric series, providing quick insights into the behavior of sums and setting the foundation for further series analysis.