Problem 74 - Precalculus: Mathematics for Calculus

Question

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$-\frac{100}{9}+\frac{10}{3}-1+\frac{3}{10}-\cdots$$

Step-by-Step Solution

Answer

The series is convergent with the sum \( -\frac{1000}{117} \).

Step 1: Identify the first term

The first term of the geometric series is given as \( a = -\frac{100}{9} \).

Step 2: Identify the common ratio

The common ratio \( r \) in an infinite geometric series can be found by dividing the second term by the first term. So, \( r = \frac{\frac{10}{3}}{-\frac{100}{9}} = -\frac{10}{3} \times \frac{9}{100} = -\frac{3}{10} \).

Step 3: Check convergence condition

For an infinite geometric series to converge, the common ratio \( r \) must satisfy \( |r| < 1 \). Check \( |-\frac{3}{10}| = \frac{3}{10} < 1 \). Thus, the series converges.

Step 4: Calculate the sum of the convergent series

If the series converges, use the formula for the sum \( S \) of an infinite geometric series: \( S = \frac{a}{1 - r} \). Substitute \( a = -\frac{100}{9} \) and \( r = -\frac{3}{10} \) into the formula:\[ S = \frac{-\frac{100}{9}}{1 + \frac{3}{10}} = \frac{-\frac{100}{9}}{\frac{13}{10}} = -\frac{100}{9} \times \frac{10}{13} = -\frac{1000}{117} \]

Key Concepts

Convergence and Divergence of Infinite Geometric Series

Understanding whether an infinite geometric series converges or diverges is pivotal. An infinite geometric series will converge if the absolute value of its common ratio, denoted as \(|r|\), is less than 1. If \(|r| < 1\), the terms of the series get progressively smaller, effectively approaching zero, which makes it possible to sum the series to a finite value. On the other hand, if \(|r| \geq 1\), the series will diverge, meaning the terms do not settle into a pattern that allows for a finite sum.

In the exercise, we encountered the common ratio \( r = -\frac{3}{10} \). Since \(|r| = \frac{3}{10}\) which is less than 1, the series converges. This is why the series can result in a specific sum instead of growing indefinitely or oscillating.

Understanding the Common Ratio

The common ratio in a geometric series is a crucial factor as it determines the nature of the series. It is found by dividing any term in the series by the term preceding it. In other words, for a series with terms \(a_1, a_2, a_3, \ldots\), the common ratio \(r\) is given by the equation \(r = \frac{a_2}{a_1}\).

For our specific series, we calculated the common ratio as \(r = -\frac{3}{10}\). This calculation involves taking the second term, \(\frac{10}{3}\), and dividing it by the first term, \(-\frac{100}{9}\). Consequently, identifying the common ratio correctly is essential, not only for determining convergence or divergence but also for calculating the sum if convergent.

Calculating the Sum of the Geometric Series

Once it is established that a geometric series converges, we can find its sum using a straightforward formula. For an infinite geometric series with a first term \(a\) and common ratio \(r\), the sum \(S\) is given by:

• \( S = \frac{a}{1 - r} \)

Given that \(a = -\frac{100}{9}\) and \(r = -\frac{3}{10}\), substituting these into the formula provides the sum. Indeed, the series from the exercise sums up as follows:

• \( S = \frac{-\frac{100}{9}}{1 + \frac{3}{10}} = -\frac{1000}{117} \)

Here, note how crucial it was to simplify the complex fraction to reach the final sum.

Identifying the First Term of a Series

The first term of a geometric series, denoted as \(a\), is foundational since it sets the pathway from which all subsequent terms are derived. It is simply the first number in the sequence that you are beginning with.

In this example, the first term is \(-\frac{100}{9}\). Revealing the first term is also vital for applying formulas for convergence and calculating sums. In the context of a problem-solving approach, correctly identifying the initial term can streamline calculations and ensure accuracy in subsequent steps.