Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use any test developed so far, including any from Section \(9.2\), to decide about the convergence or divergence of the series. Give a reason for your conclusion. $$ \sum_{k=1}^{\infty}\left(\frac{3}{\pi}\right)^{k} $$

Step-by-Step Solution

Answer

The series converges because \( |r| = \left|\frac{3}{\pi}\right| < 1 \).

Step 1: Identify the Type of Series

The given series is \( \sum_{k=1}^{\infty} \left(\frac{3}{\pi}\right)^{k} \), which is a geometric series of the form \( \sum_{k=1}^{\infty} ar^k \) with \( a = \left(\frac{3}{\pi}\right) \) and \( r = \left(\frac{3}{\pi}\right) \).

Step 2: Recall the Convergence Criterion for a Geometric Series

A geometric series \( \sum_{k=0}^{\infty} ar^k \) converges if the absolute value of the common ratio \( r \) is less than 1, i.e., \( |r| < 1 \). If \( |r| \geq 1 \), the series diverges.

Step 3: Evaluate the Common Ratio

Calculate the absolute value of the common ratio \( r = \frac{3}{\pi} \). Since \( \pi \approx 3.14 \), \( \frac{3}{\pi} \approx 0.955 < 1 \). Therefore, \( |r| = \left|\frac{3}{\pi}\right| < 1 \).

Step 4: Conclusion Based on Convergence Criterion

Since the absolute value of the common ratio \( \left|\frac{3}{\pi}\right| < 1 \), the series \( \sum_{k=1}^{\infty} \left(\frac{3}{\pi}\right)^{k} \) converges by the geometric series test.

Key Concepts

Understanding Convergence in Geometric Series

Convergence is an important concept when dealing with sequences and series, particularly with geometric series. A series converges if the sum of its terms approaches a specific value as more terms are added. For a geometric series, this depends heavily on the common ratio, denoted as \( r \). If \(|r| < 1\), the series converges, meaning it sums to a finite number. Otherwise, the series diverges, growing indefinitely without reaching a finite limit.
If we refer to the exercise, we identify our series as \( \sum_{k=1}^{\infty}\left(\frac{3}{\pi}\right)^{k} \). Here, \( a \) (the first term) and the common ratio \( r \) are both \( \frac{3}{\pi} \). Calculating the common ratio gives \( \left|\frac{3}{\pi}\right| \approx 0.955 \), which is less than 1.
This means our series converges. When you learn to determine convergence, it helps you understand if, and how, an infinite number of terms can sum to a particular value. The conditions \(|r| < 1\) for convergence is a critical foundational rule when studying geometric series.

Sequence and Series Explained

The distinction between sequences and series is essential. A sequence is a list of numbers written in a specific order. A series, meanwhile, is the sum of the terms of a sequence. In mathematics, sequences can be finite or infinite, leading to finite or infinite series.
Geometric sequences and series have a simple structure where each term after the first is found by multiplying the previous one by a constant called the common ratio \( r \). This simplicity makes geometric series very popular in mathematical studies and applications.
In the exercise, the series \( \sum_{k=1}^{\infty}\left(\frac{3}{\pi}\right)^{k} \) illustrates this concept with an infinite number of terms. By analyzing such a series, you see how each additional term contributes to the overall sum. This method is a central part of accurately calculating and understanding infinite series.

Basics of Mathematical Proofs: Convergence Criterion

Mathematical proofs are logical arguments ensuring that statements, such as those regarding convergence, hold true. They are central to validating mathematical concepts and assertions. Regarding the convergence of a series, a common proof involves using well-established tests like the geometric series test.

• The basic principle of mathematical proofs is to present logical steps that lead to a conclusion. For geometric series, we use the test of \(|r| < 1\) to demonstrate convergence.
• In the given series \( \sum_{k=1}^{\infty}\left(\frac{3}{\pi}\right)^{k} \), our proof involves calculating \( |r| \) and verifying that it's less than 1, thereby confirming convergence.

This method provides a clear, concise argument or "proof" that the statement regarding the series' convergence is correct. By mastering mathematical proofs, you gain confidence in solving problems and verifying solutions independently, reinforcing the learning process.