Problem 18 - Calculus: Early Transcendentals

Question

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( 4 + 3 + \frac {9}{4} + \frac {27}{16} + \cdot \cdot \cdot \)

Step-by-Step Solution

Answer

The series is convergent and its sum is 16.

Step 1: Identify the First Term

The first step in solving whether a geometric series converges or diverges is to identify the first term of the series. In this series, the first term \(a\) is 4.

Step 2: Identify the Common Ratio

The next step is to determine the common ratio \(r\) of the geometric series. We calculate the common ratio by dividing the second term by the first term, i.e., \( r = \frac{3}{4} \).

Step 3: Determine Convergence or Divergence

A geometric series converges if the absolute value of the common ratio \( |r| \) is less than 1. Since \( |\frac{3}{4}| = 0.75 < 1 \), the series is convergent.

Step 4: Calculate the Sum of the Convergent Series

The sum of an infinite convergent geometric series is given by the formula \( S = \frac{a}{1-r} \). Substitute \( a = 4 \) and \( r = \frac{3}{4} \) into the formula to get \( S = \frac{4}{1-\frac{3}{4}} = \frac{4}{\frac{1}{4}} = 16 \).

Key Concepts

Convergence and Divergence

When studying geometric series, an important aspect to understand is the concept of convergence and divergence. These terms describe the behavior of the series as the number of terms approaches infinity.

• Convergence: A series converges if the sum of its terms approaches a finite number as the number of terms increases. In a geometric series, convergence is determined by the absolute value of the common ratio, \(|r|\). If \(|r| < 1\), the series will converge.
• Divergence: Conversely, a series diverges if the sum does not settle towards a particular number but instead grows indefinitely. For a geometric series, if \(|r| \geq 1\), the series will diverge.

To determine convergence, follow these steps:1. Identify the common ratio, \(r\).2. Check if \(|r| < 1\) for convergence.For the exercise provided, we identified a common ratio of \(\frac{3}{4}\), implying convergence since \(0.75 < 1\). The series is convergent.

Infinite Series

An infinite series is a summation that continues indefinitely. This kind of series can exhibit both convergent and divergent behaviors based on its terms and structure.

In an infinite geometric series, the terms decrease (or increase) based on a consistent pattern, dictated by the common ratio. This predictable change in term values makes analyzing geometric series very mathematical.

Considerations for infinite series include:

• Infinite Nature: The series does not have an ending term; instead, it approaches an ultimate value or behavior as more terms are added.
• Summation of Terms: If the series converges, we can find a finite sum that describes its total, despite having endless terms.

For the geometric series in the exercise, the infinite nature was handled by using the convergence criterion and summation formula, helping us to find a definite value for the series despite its infinite terms.

Sum of Geometric Series

Calculating the sum of an infinite geometric series requires a specific approach since simply adding all the terms infinitely is impractical.

In a convergent geometric series, the sum is obtained using the formula:\[ S = \frac{a}{1-r} \]This formula works only when the series converges, i.e., \(|r| < 1\). Let's breakdown the formula:

• \(a\) is the first term of the series.
• \(r\) is the common ratio between consecutive terms.
• \(S\) represents the sum towards which the series approaches.

In the exercise, after confirming that the series converges with \(r = \frac{3}{4}\), we applied the sum formula:\[ S = \frac{4}{1 - \frac{3}{4}} = 16 \]Thus, the infinite series adds up to a finite sum of 16.