Problem 43 - A Graphical Approach to Precalculus with Limits

Question

CONCEPT CHECK Under what conditions does the sum of the terms of an infinite geometric sequence exist?

Step-by-Step Solution

Answer

The sum exists if the common ratio \( -1 < r < 1 \).

Step 1: Understanding Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted as \( r \).

Step 2: Defining Infinite Geometric Series

An infinite geometric series is the sum of the terms of an infinite geometric sequence. It can be expressed as \( a + ar + ar^2 + ar^3 + \, ... \, \) where \( a \) is the first term and \( r \) is the common ratio.

Step 3: Condition for Convergence

For the sum of an infinite geometric series to exist (i.e., for the series to converge), the common ratio \( r \) must satisfy \( -1 < r < 1 \). This condition ensures that the terms of the series become smaller as we sum more terms, allowing the total sum to approach a finite value.

Step 4: Determining the Sum Formula

If \( |r| < 1 \), the sum of the infinite geometric series can be calculated using the formula: \( S = \frac{a}{1 - r} \), where \( S \) is the sum of the series, \( a \) is the first term, and \( r \) is the common ratio.

Key Concepts

Convergence of Series

In mathematics, the concept of convergence is vital to understanding infinite series, particularly infinite geometric series. When we discuss convergence, we are essentially examining whether the sum of an infinite series approaches a finite value as we continue to sum more terms.
For an infinite geometric series, convergence depends on the common ratio, denoted as \( r \). Specifically, the series will converge if and only if the common ratio \( r \) falls within the range \(-1 < r < 1\).
This condition ensures that each subsequent term in the series becomes progressively smaller, allowing the sum to gradually approach a fixed finite value, rather than growing indefinitely. When a series meets these criteria, we say that it "converges" to a sum.

Geometric Sequences

A geometric sequence is defined by its method of construction: start with any number, called the first term \( a \), and repeatedly multiply by a constant known as the common ratio \( r \). For example, in a sequence like 2, 6, 18, 54, ... each term is obtained by multiplying the previous term by 3.
Geometric sequences have a beautiful property of scalability, which means operations like scaling or shrinking uniformly modify the entire sequence. This characteristic becomes useful in various real-world scenarios, from calculating compound interest to understanding population growth patterns.
Recognizing and constructing these sequences by identifying the first term and the common ratio is the key to unlocking their full potential in arithmetic and beyond.

Sum Formula for Geometric Series

When analyzing a geometric series, especially infinite ones, being able to determine the sum is powerful. The sum formula for an infinite geometric series, when it converges, provides a straightforward way to calculate the total with just a few parameters.
If the common ratio \( r \) of a series is such that \(|r| < 1\), the sum \( S \) of the series can be determined using the formula \( S = \frac{a}{1 - r} \). Here, \( a \) represents the first term, and the formula elegantly sums the infinite number of terms to give a finite result.
Understanding this formula not only makes solving problems much quicker but also provides insights into the nature of geometric growth and decay across infinite terms.