Problem 17 - Thomas Calculus

Question

In Exercises \(15-18,\) determine if the geometric series converges or diverges. If a series converges, find its sum. $$ \left(\frac{1}{8}\right)+\left(\frac{1}{8}\right)^{2}+\left(\frac{1}{8}\right)^{3}+\left(\frac{1}{8}\right)^{4}+\left(\frac{1}{8}\right)^{5}+\cdots $$

Step-by-Step Solution

Answer

The series converges, and its sum is \( \frac{1}{7} \).

Step 1: Identify the First Term and Common Ratio

A geometric series takes the form \( a + ar + ar^2 + ar^3 + \cdots \). Here, the first term \( a \) is \( \frac{1}{8} \), and the common ratio \( r \) is also \( \frac{1}{8} \).

Step 2: Check Convergence Criteria

For a geometric series \( a + ar + ar^2 + \cdots \) to converge, the absolute value of the common ratio \( |r| \) must be less than 1. In this problem, \( r = \frac{1}{8} \), and since \( \left| \frac{1}{8} \right| < 1 \), the series converges.

Step 3: Use the Convergence Formula for the Sum

When a geometric series converges, its sum \( S \) is given by the formula \( S = \frac{a}{1 - r} \). Substitute \( a = \frac{1}{8} \) and \( r = \frac{1}{8} \) into this formula: \[ S = \frac{\frac{1}{8}}{1 - \frac{1}{8}}. \]

Step 4: Simplify the Expression for the Sum

Calculate the denominator: \( 1 - \frac{1}{8} = \frac{8}{8} - \frac{1}{8} = \frac{7}{8} \). Substitute back into the formula to find \( S \): \[ S = \frac{\frac{1}{8}}{\frac{7}{8}} = \frac{1}{8} \times \frac{8}{7} = \frac{1}{7}. \]

Key Concepts

Convergence

In the context of a geometric series, convergence refers to whether the sum of the series reaches a finite value as the number of terms increases indefinitely. When a series converges, this is interesting because it allows you to find the sum of what appears to be an infinite sequence of numbers.

To determine convergence, examine the common ratio, often represented by "r". For a series to converge, the absolute value \(|r|\) must be less than 1. This ensures that as more terms are added, each subsequent term shrinks in size, causing the sum to reach a specific, finite number.

In our exercise, with a common ratio of \(\frac{1}{8}\), convergence is confirmed since \(\left| \frac{1}{8} \right| < 1\). This tells us the series will add up to a finite sum eventually.

Sum of Series

When a geometric series is verified to converge, the sum of the series is not only finite but can also be calculated straightforwardly using a formula.

The formula is: \[ S = \frac{a}{1 - r} \] where \(a\) represents the first term, and \(r\) is the common ratio. Plugging these values into the formula provides the exact sum of the entire series, despite it theoretically having infinite terms.

As seen in the exercise, substituting \(a = \frac{1}{8}\) and \(r = \frac{1}{8}\) into the formula gives us a sum \(S\) of \(\frac{1}{7}\). This means, remarkably, that if you added up each term of this infinite sequence, you would end up with \(\frac{1}{7}\).

Common Ratio

The common ratio is a key factor in determining the behavior of a geometric series. It is the factor by which you multiply each term to get to the next one. You compute it by dividing any term in the series by the one just before it.

In our series, the common ratio is \(\frac{1}{8}\). This means every term is \(\frac{1}{8}\) times the previous term. It functions like the decline rate of the series, as each term becomes progressively smaller if the ratio is a fraction between 0 and 1

A common ratio tells a lot about the series' characteristics. If it were larger than 1, the series would diverge because the terms would get increasingly larger. If it were exactly 1, each term would be the same, and the series wouldn't reach a particular sum.

First Term

The first term in a geometric series, denoted as \(a\), is essential as it is the building block from which the entire series grows.

In the example given, the first term \( a = \frac{1}{8} \) sets the stage for every subsequent calculation. It is the starting point for the series and is used in combination with the common ratio to determine the behavior and sum of the series.

Knowing the first term helps complete the formula for calculating the series' sum. Altering the first term would change the sum drastically, confirming the importance of knowing both the first term and the common ratio when analyzing a geometric series.