Problem 49 - College Algebra with Modeling and Visualization

Question

Find the sum of the infinite geometric series. $$ 1-\frac{1}{10}+\frac{1}{100}-\frac{1}{1000}+\dots+\left(-\frac{1}{10}\right)^{n-1}+\cdots $$

Step-by-Step Solution

Answer

The sum of the series is \( \frac{10}{11} \).

Step 1: Identify First Term

The first term of the geometric series is given directly in the sequence, which is 1. This is the value of the term when \( n = 1 \).

Step 2: Determine Common Ratio

Observe how each term relates to the previous one. Start with the first few terms: 1, -1/10, 1/100, -1/1000. The common ratio \( r \) is the factor you multiply a term by to get the next term. Here, each term is multiplied by \(-1/10\), so \( r = -1/10 \).

Step 3: Confirm Series is Infinite and Geometric

Verify the series is infinite and geometric. Since the terms continue indefinitely and have a constant common ratio \( r = -1/10 \), the series is indeed infinite and geometric.

Step 4: Use Sum Formula for Infinite Geometric Series

The formula for the sum \( S \) of an infinite geometric series where \(|r| < 1\) is \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio.

Step 5: Calculate the Sum

Substitute \( a = 1 \) and \( r = -1/10 \) into the sum formula: \[ S = \frac{1}{1 - (-1/10)} = \frac{1}{1 + 1/10} = \frac{1}{1.1} = \frac{10}{11}. \]

Step 6: Interpret Result

The calculation yields a sum \( S = \frac{10}{11} \). This means the sum of the infinite geometric series \( 1 - \frac{1}{10} + \frac{1}{100} - \frac{1}{1000} + \cdots \) converges to the value \( \frac{10}{11} \).

Key Concepts

Understanding Geometric Sequences

Geometric sequences are fascinating as they involve numbers that follow a specific pattern dictated by a constant called the common ratio. In a geometric sequence, each term is obtained by multiplying the previous term by this common ratio. For example, in the sequence 1, \(-\frac{1}{10}\), \(\frac{1}{100}\), \(-\frac{1}{1000}\), and so on, we can deduce that the common ratio \(r\) is \(-\frac{1}{10}\). This pattern continues, creating a sequence that can either increase or decrease based on the value of the common ratio.

Key characteristics of geometric sequences include:

• Constant multiplication (or division for fractions) across the sequence.
• If \(|r| < 1\), the terms decrease towards zero.
• If \(r > 1\) or \(r < -1\), the terms grow in value or alternate between huge and tiny due to sign changes.

These sequences are fundamental in understanding more complex concepts like series and convergence.

Exploring Infinite Series

When a sequence is extended indefinitely, it forms what is known as an infinite series. With a geometric sequence, the series can be written by adding all terms of the sequence together. The sequence becomes a series when written as a sum: \(1 - \frac{1}{10} + \frac{1}{100} - \frac{1}{1000} + \cdots\).

Understanding an infinite series involves looking at how its sum behaves as we add more terms:

• If the terms keep decreasing and approach zero, the series might have a finite sum.
• If the terms don’t approach zero, the series could diverge, meaning it doesn't settle to a fixed value.

Infinite series are powerful in mathematical analysis, providing ways to express functions and even solve real-world problems.

Series Convergence and its Implications

Series convergence is a critical concept in understanding infinite series. For a better perspective, consider the infinite geometric series \(1 - \frac{1}{10} + \frac{1}{100} - \frac{1}{1000} + \cdots\). This series converges because the sum of its infinite terms approaches a fixed number, specifically \(\frac{10}{11}\).

Convergence is determined by the common ratio \(r\) of a geometric series where \(|r| < 1\). The series converges, and the sum \(S\) can be calculated using the formula:

• \(S = \frac{a}{1 - r}\)
• where \(a\) is the first term.

This formula is only applicable when \(|r| < 1\) because:

• If \(|r| \geq 1\), the series diverges, meaning it extends towards infinity or negative infinity without reaching a stable sum.

The concepts of convergence help in numerous applications, from evaluating financial investments to analyzing algorithms in computer science.