Problem 9 - Essential Calculus Early Transcendentals

Question

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{4^{n}}$$

Step-by-Step Solution

Answer

The series is convergent, and its sum is \( \frac{4}{7} \).

Step 1: Identify the Common Ratio

The given series is \( \sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{4^{n}} \). By writing the first few terms, we have \( \frac{1}{4} - \frac{3}{16} + \frac{9}{64} - \frac{27}{256} + \ldots \). We see that the series is of the form \( a r^{n-1} \) where \( a \) is the first term, and \( r = \frac{-3}{4} \) is the common ratio.

Step 2: Determine Convergence

A geometric series \( \sum a r^{n-1} \) is convergent if the absolute value of the common ratio \( |r| \) is less than 1. Here, the common ratio \( r = \frac{-3}{4} \) has \( |r| = \frac{3}{4} < 1 \). Thus, the series is convergent.

Step 3: Find the Sum of the Series

For a convergent geometric series \( \sum a r^{n-1} \), the sum \( S \) can be found using the formula \( S = \frac{a}{1-r} \). Here, \( a = 1 \) and \( r = \frac{-3}{4} \). Therefore, the sum is \[ S = \frac{1}{1 - \left( \frac{-3}{4} \right)} = \frac{1}{1 + \frac{3}{4}} = \frac{1}{\frac{7}{4}} = \frac{4}{7}. \]

Key Concepts

Convergence of Series

A fundamental concept in understanding geometric series is the evaluation of its convergence. Convergence, in the context of series, refers to whether a series sums up to a finite number as more terms are added. This is a crucial aspect because it determines whether the series leads to a meaningful result or not.

In geometric series, convergence is principally determined by the common ratio \(r\), signifying the factor by which each term is multiplied to yield the next term. For a geometric series to be convergent, the absolute value of the common ratio \(|r|\) must be less than 1. Why is this important?

• If \(|r| < 1\), each subsequent term in the series becomes smaller, leading the series towards a finite limit.
• Conversely, if \(|r| \geq 1\), the terms do not diminish as significantly, often causing the series to diverge, or not reach a specific finite sum.

This insight can be critical for various mathematical applications, from solving problems to applying these concepts in real-world scenarios, like financial modeling or physics.

Common Ratio

Understanding the common ratio is key to demystifying geometric series. The common ratio \(r\) is the constant factor that relates consecutive terms of the series. In other words, to get from one term to the next, you multiply by \(r\).

In the given series, represented as \( \sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{4^{n}} \), the common ratio \(r\) can be determined by observing how the sign and the magnitude change from one term to another.

• For this series, the common ratio is \(r = \frac{-3}{4}\), which means each term is \( \frac{-3}{4} \) times the preceding term.
• The initial term \(a\), which is always crucial in these calculations, is 1 when \(n=1\).
• The sign of \(r\) indicates that the sequence alternates, meaning it may switch between positive and negative values.

Having identified \(r\), we have a clearer view on whether the series will converge and how to calculate further attributes such as the sum of the series.

Sum of Series

Once the convergence of a geometric series is established (\(|r| < 1\)), calculating the sum becomes possible. This sum reflects the total value towards which the series converges.

For a convergent geometric series, the formula for the sum \(S\) is:\[S = \frac{a}{1 - r}\]
Where:

• \(a\) is the first term of the series.
• \(r\) is the common ratio.

In the context of the exercise, the first term \(a\) is 1, and the common ratio \(r\) is \(-\frac{3}{4}\). So, applying these:\[S = \frac{1}{1 - \left(-\frac{3}{4}\right)} = \frac{1}{1 + \frac{3}{4}} = \frac{1}{\frac{7}{4}} = \frac{4}{7}\]
This calculation shows that even when adding an infinite number of alternating terms, the sum converges to a finite value of \( \frac{4}{7} \). This demonstrates the incredible power of series in reaching beyond simple addition into a realm of precise, finite calculations.