Problem 5 - Algebra and Trigonometry Real Mathematics, Real People

Question

For what values of the common ratio \(r\) is it possible to find the sum of an infinite geometric series?

Step-by-Step Solution

Answer

The values of \( r \) for which the sum of an infinite geometric series can be calculated are real numbers lying in the open interval (-1,1).

Step 1: Understand the Concept of Geometric Series

A geometric series 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. We represent an infinite geometric series as \( a + ar + ar^2 + ar^3 + . . . \) where \( a \) is the first term and \( r \) is the common ratio.

Step 2: Explain the Sum of an Infinite Geometric Series

The sum \( S \) of an infinite geometric series can be calculated using the formula \( S = \frac{a}{1 - r} \) where \( r \) is the common ratio and \( a \) is the first term. This formula is valid only if the absolute value of \( r \) is less than 1, i.e., \( -1 < r < 1 \). This condition ensures that the terms in the series become smaller and smaller, enabling a finite sum.

Step 3: Determine The Range of Common Ratio

From the earlier step, it's clear that the sum of an infinite geometric series is calculable only when the absolute value of the common ratio \( r \) is less than 1. Therefore, the possible values of \( r \) are all real numbers within the open interval (-1,1).

Key Concepts

Common Ratio

The concept of a common ratio is central to understanding geometric sequences and series. In a geometric sequence, each term is derived by multiplying the previous term by a constant factor. This constant factor is what we call the common ratio. It remains the same throughout the sequence.

• If the common ratio is greater than 1, each term becomes larger.
• If the common ratio is between 0 and 1, each term gets progressively smaller.
• A negative common ratio will cause terms to alternate in sign.

Understanding the common ratio helps us predict how the sequence behaves as it progresses. For example, a negative ratio with an odd term count will end with a negative term.

Sum of Geometric Series

The sum of an infinite geometric series allows us to calculate the total of all terms in a series when continued indefinitely. The formula used is \[ S = \frac{a}{1 - r} \]where:

• \( S \) is the sum of the series,
• \( a \) is the first term, and
• \( r \) is the common ratio.

This formula applies when the series has an infinite number of terms and a common ratio where the absolute value is less than one. If these conditions are met, the terms become smaller, making it possible to sum them to a finite number.

Geometric Sequence

A geometric sequence is a series of numbers where each term is obtained by multiplying the preceding one by a constant, known as the common ratio. The general form is:\[ a, ar, ar^2, ar^3, \ldots \]where:

• \( a \) is the initial term,
• \( r \) is the common ratio,
• Each term is a product of the previous term and \( r \).

Geometric sequences can converge or diverge based on the common ratio. These sequences are foundational in mathematics due to their simple yet profound implications in various fields, such as finance and physics.

Convergence Conditions

The convergence of an infinite geometric series is a key aspect in being able to calculate its sum. For an infinite geometric series to converge and have a sum, the common ratio must satisfy the condition \(-1 < r < 1\). This condition implies that:

• Each succeeding term diminishes in size as the sequence progresses.
• The terms "shrink" towards zero, leading to a calculable constant sum.

If these conditions are not met, the series diverges, meaning it continues to grow indefinitely and cannot have a finite sum. Therefore, the ratio ensures that the series doesn't grow out of bounds.