Problem 3 - Algebra and Trigonometry Real Mathematics, Real People

Question

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 an 'infinite geometric series'.

Step 1: Understanding Infinite Geometric Series

An infinite geometric series is the sum of the terms in an infinite geometric sequence. 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.

Step 2: Identifying the term for the sum of an infinite geometric sequence

The sum of terms in an infinite geometric sequence is called a series. If the common ratio is between -1 and 1, this series converges and we can calculate its sum, otherwise it diverges. The term for the sum of terms in an infinite geometric sequence is called an 'infinite geometric series'.

Key Concepts

Understanding Geometric Sequence

A geometric sequence is an ordered list of numbers where each number, called a term, is determined by multiplying the previous term by a constant value. This constant value is known as the common ratio. To find the second term from the first, or the third from the second, simply multiply by this ratio.

• For example, in the sequence 2, 6, 18, 54, ... the common ratio is 3.
• Multiply 2 by 3 to get 6, multiply 6 by 3 to get 18, and so on.

This type of sequence allows us to quickly generate terms and easily determine large numbers within the sequence. Understanding geometric sequences provides the foundation for comprehending related mathematical concepts like series.

Defining the Common Ratio

The common ratio is a crucial element in defining a geometric sequence. It dictates how the sequence evolves and can dramatically influence the series formed from the sequence. Given any geometric sequence, the common ratio (often denoted as \( r \)) is found by dividing any term by its preceding term.

• If the sequence is 5, 15, 45, 135, ... the common ratio is \( r = \frac{15}{5} = 3 \).
• The ratio can be positive, negative, fractional or whole.

The value of the common ratio affects whether a geometric series will converge or diverge. This is crucial for infinite series as it determines if we can calculate the sum.

Exploring Series Convergence

Series convergence refers to the behavior of an infinite series in terms of reaching a fixed sum as more terms are added. For an infinite geometric series to converge, the absolute value of its common ratio \( r \) must be less than 1, specifically \( -1 < r < 1 \). This condition ensures that as you add more and more terms from the geometric sequence, the terms get smaller and the total approaches a certain value, instead of heading off to infinity.

• For example, in the series \( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \) the common ratio \( r = \frac{1}{2} \) and the series converges.
• Series that do not meet this condition with \(|r| \geq 1\) will diverge, meaning they do not approach a finite sum.

By understanding series convergence, students can determine whether it is possible to find a sum for an infinite series and apply this to practical calculations.