Problem 9 - Thomas Calculus

Question

In Exercises \(7-14,\) write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. $$\sum_{n=1}^{\infty}\left(1-\frac{7}{4^{n}}\right)$$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Understanding the Series

The series given is \( \sum_{n=1}^{\infty} \left(1 - \frac{7}{4^n}\right) \). This is an infinite series where each term is \(1 - \frac{7}{4^n}\). Our task is to write out the first eight terms and then determine if the series converges or diverges.

Step 2: Writing the First Eight Terms

To find the first eight terms, substitute \(n = 1, 2, 3, ..., 8\) into the formula for the general term: - For \(n = 1\), the term is \(1 - \frac{7}{4} = 1 - 1.75 = -0.75\).- For \(n = 2\), the term is \(1 - \frac{7}{16} = 1 - 0.4375 = 0.5625\).- For \(n = 3\), the term is \(1 - \frac{7}{64} = 1 - 0.109375 = 0.890625\).- For \(n = 4\), the term is \(1 - \frac{7}{256} = 1 - 0.02734375 = 0.97265625\).- For \(n = 5\), the term is \(1 - \frac{7}{1024} = 1 - 0.0068359375 = 0.9931640625\).- For \(n = 6\), the term is \(1 - \frac{7}{4096} = 1 - 0.001708984375 = 0.998291015625\).- For \(n = 7\), the term is \(1 - \frac{7}{16384} = 1 - 0.00042724609375 = 0.99957275390625\).- For \(n = 8\), the term is \(1 - \frac{7}{65536} = 1 - 0.0001068115234375 = 0.9998931884765625\).

Step 3: Observing the Pattern

From the first few terms, it is evident that each term is converging to 1 as \(n\) increases. This suggests that the series may converge towards a certain sum rather than diverge.

Step 4: Analyzing the Series

The series can be split into two separate series: 1. \(\sum_{n=1}^{\infty} 1 = \infty\), which diverges.2. \(-\sum_{n=1}^{\infty} \frac{7}{4^n}\), which is a geometric series with first term \(-\frac{7}{4}\) and ratio \(\frac{1}{4}\).The term \(-\sum_{n=1}^{\infty} \frac{7}{4^n}\) is a convergent geometric series since \(|r| < 1\). This series sum can be calculated as the first term divided by \(1 - r\): \[-\sum_{n=1}^{\infty} \frac{7}{4^n} = \frac{-\frac{7}{4}}{1 - \frac{1}{4}} = \frac{-\frac{7}{4}}{\frac{3}{4}} = -\frac{7}{3}\].

Step 5: Determining the Sum of the Series

Overall, the series \(\sum_{n=1}^{\infty}\left(1 - \frac{7}{4^n}\right)\) diverges because while the geometric series part \( -\frac{7}{4^n}\) converges to \(-\frac{7}{3}\), the constant term \(\sum_{n=1}^{\infty}1\) diverges to infinity. Thus, the entire series diverges.

Key Concepts

Infinite Series

An infinite series is a sum of infinitely many terms. It is written in the form \( \sum_{n=1}^{\infty} a_n \), where each \( a_n \) is a term in the series. In simpler terms, it means adding together an infinite number of terms in a sequential manner. Infinite series are crucial in mathematics because they allow us to express complex ideas in a sum of simpler terms.
They come in many forms and can be either convergent or divergent. Understanding whether an infinite series converges or diverges plays a significant role in determining the behavior and value of the series.

Convergence and Divergence

Convergence and divergence are key concepts when dealing with infinite series. A series converges if the sum of its terms approaches a specific, finite number as you continue to add more terms. Think of it as getting closer and closer to a particular value.
On the other hand, a series diverges if the sum does not approach any finite value. This could mean that the series grows without bounds or oscillates indefinitely without settling down.

• A convergent series has a limit to which the terms of the series approach as they are added.
• A divergent series does not settle at a particular value or grows infinitely.

Understanding convergence and divergence helps in determining the behavior and potential outcomes of the series.

Geometric Series Sum

Geometric series are a specific type of series where each term is a constant multiple (called the ratio) of the previous term. A typical geometric series looks like \( a + ar + ar^2 + ar^3 + \ldots \), where \( a \) is the first term and \( r \) is the common ratio.
The sum of an infinite geometric series converges only when the absolute value of the ratio \(|r| < 1\). Otherwise, the series will diverge. The formula for the sum \( S \) of an infinite geometric series where \(|r| < 1\) is:\[S = \frac{a}{1 - r}\]If the series is convergent, this formula provides a simple way to calculate the sum of the series, making it very useful for problems involving geometric series.

Geometric Series Ratio

In a geometric series, the ratio \( r \) is the factor by which each subsequent term is multiplied to get the next term in the series.
This ratio is critical for determining the behavior of the series. Whether a geometric series converges or diverges depends on this ratio:

• If \(|r| < 1\), the series converges.
• If \(|r| \geq 1\), the series diverges.

Analyzing the ratio of a geometric series allows us to quickly predict whether its sum approaches a finite value, which is essential for solving complex series-related problems. The concept of the geometric ratio simplifies the evaluation process and plays a central role in understanding geometric series.