Problem 21 - Calculus: Early Transcendentals

Question

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} 12(0.73)^{n-1} \)

Step-by-Step Solution

Answer

The series is convergent with a sum of approximately 44.44.

Step 1: Identify the First Term and Common Ratio

The given geometric series is \( \sum_{n = 1}^{\infty} 12(0.73)^{n-1} \). This is in the form \( \sum_{n = 1}^{\infty} ar^{n-1} \), where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = 12 \) and \( r = 0.73 \).

Step 2: Determine Convergence

A geometric series converges if the absolute value of the common ratio \( r \) is less than 1, i.e., \( |r| < 1 \). Since \( |0.73| = 0.73 < 1 \), the series is convergent.

Step 3: Calculate the Sum of the Series

The sum \( S \) of an infinite convergent geometric series \( \sum_{n = 1}^{\infty} ar^{n-1} \) is given by \( S = \frac{a}{1-r} \). Substituting the values \( a = 12 \) and \( r = 0.73 \), we find \( S = \frac{12}{1-0.73} = \frac{12}{0.27} \approx 44.44 \).

Key Concepts

Convergence of Series

Convergence in the context of a series is an essential understanding for students of mathematics. When we talk about the convergence of a series, essentially we refer to whether the infinite sum results in a finite number. This is particularly important when dealing with geometric series.

A geometric series will converge if the absolute value of the common ratio is less than 1. Mathematically, this is expressed as \(|r| < 1\). This condition arises because if the common ratio is greater than or equal to 1 in absolute value, the terms of the series do not approach zero, causing the sum to diverge.

In the original exercise, the series \(\sum_{n = 1}^{\infty} 12(0.73)^{n-1}\) was checked for convergence by confirming that \(|0.73| = 0.73 < 1\). This indicates convergence. Being able to identify whether a series converges allows students to know if their calculations of total sums will be finite and meaningful.

Infinite Series

An infinite series is a series that extends indefinitely. In other words, it is a sum of an infinite sequence of terms. Examples include geometric series, arithmetic series, and others. Understanding infinite series is crucial because they model many phenomena in mathematics, physics, engineering, and finance.

In the case of a geometric series, which is a series where each term is a constant ratio of the previous term, the analysis often involves determining the behavior as more terms are included. However, adding an infinite number of terms can still yield a finite sum if the series converges. This is the case for our exercise, where the series is infinite but convergent through the analysis of its common ratio.

• A common ratio \(|r| < 1\) leads to convergence.
• A common ratio \(|r| \geq 1\) leads to divergence.

Understanding infinite series helps students grasp the kinds of infinities they're working with, and when calculations can actually resolve to meaningful numbers.

Sum of Series

Calculating the sum of a convergent geometric series can seem like magic as it provides a finite result despite the infinite addition process. The sum formula for a geometric series \( \sum_{n=1}^{\infty} ar^{n-1} \) is given as \( S = \frac{a}{1-r} \), where \(a\) is the first term, and \(r\) is the common ratio.

This formula works based on the fact that each subsequent term gets increasingly smaller, approaching zero, which effectively closes off the series in a bounded manner. In our example, the first term \(a\) is 12 and the common ratio \(r\) is 0.73, leading to a sum:\[ S = \frac{12}{1-0.73} = \frac{12}{0.27} \approx 44.44 \]This calculation provides the total value of all terms of the series added together infinitely.

For learners, this formula is incredibly powerful. It shows that infinity does not always mean impossibility in calculations and can be tackled with the right mathematical tools.