Problem 4 - Precalculus with Limits

Question

Fill in the blanks. The sum of the terms of an infinite geometric sequence is called a ________ ________.

Step-by-Step Solution

Answer

The sum of the terms of an infinite geometric sequence is called a 'geometric series'.

Step 1: Understanding an Infinite Geometric Sequence

An infinite 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.

Step 2: Understanding the Sum of an Infinite Geometric Sequence

The sum of the terms of an infinite geometric sequence, when exists, is obtained by the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. It is important to note that the sum only exists when the absolute value of \( r \) is less than 1, i.e., \( |r| < 1 \).

Step 3: Identifying the Missing Term

With this understanding, we can fill in the blank. The term used to describe the sum of the terms of an infinite geometric sequence is 'geometric series'.

Key Concepts

Geometric Sequence

A geometric sequence is a string of numbers where each term, after the first, is the result of multiplying the previous term by a constant value. This constant is known as the common ratio. For example, in the sequence 2, 6, 18, 54, ..., each term is obtained by multiplying the previous term by 3. The sequence keeps growing indefinitely, making it infinitely long.

A key property of a geometric sequence is this regular pattern of multiplication, which remains consistent throughout the sequence. The geometric sequence is formally expressed as:

• The first term: \( a \)
• The common ratio: \( r \)
• The nth term: \( a \times r^{(n-1)} \)

Understanding geometric sequences is fundamental, as it sets the stage for learning about the convergence and the sum of infinite series.

Common Ratio

The common ratio is the backbone of the geometric sequence. It is the factor by which consecutive terms of the sequence are obtained. Represented as \( r \) in equations, it is a constant value that can be positive, negative, or even a fraction.

When \( r > 1 \), the terms in the sequence grow larger and getting exponentially bigger. If \( r < 1 \) and positive, the terms decrease but never reach zero. A negative \( r \) causes the terms to alternate between positive and negative.

Understanding the role of the common ratio is crucial because it affects whether the sum of the sequence converges to a limit or diverges. A geometric sequence with \( |r| < 1 \) has a sum that can be calculated using specific formulas, which is not the case if the absolute value of \( r \) is equal to or greater than 1.

Convergence of Series

The concept of convergence is vital when discussing infinite geometric sequences. A series converges if the sum of its infinitely many terms approaches a finite number. Otherwise, it diverges.

For an infinite geometric series, convergence is determined by the common ratio. If the common ratio \( |r| < 1 \), the series converges and you can find its sum using the formula \( S = \frac{a}{1 - r} \), where \( 'a' \) is the first term. However, if \( |r| \geq 1 \), the series does not approach a limit and thus diverges.

This criterion for convergence is essential for understanding the behavior of geometric series as it gives insight into whether the infinite sum of the series is a finite, meaningful number or not. The idea of convergence not only applies to simple sequences but also plays a major role in higher forms of mathematics, such as the study of power series and function approximations.