Question

Explain how to tell whether the series \(\sum_{i=1}^{\infty} a_1 r^{i-1}\) has a sum.

Short answer

The series \(\sum_{i=1}^{\infty} a_1 r^{i-1}\) has a sum only if the absolute value of the common ratio \(r\) is less than 1. If the absolute value of \(r\) is less than 1, the series is said to be convergent and has a sum. Otherwise, if the absolute ratio of \(r\) is equal to or greater than 1, the series is divergent and does not have a sum.

Step-by-step solution

Identify the parts of the geometric series

A geometric series has the form \(\sum_{i=1}^{\infty} a r^{i-1}\), where \(a\) is the first term and \(r\) is the common ratio. In this case, the given series is \(\sum_{i=1}^{\infty} a_1 r^{i-1}\), in which \(a_1\) is the first term and \(r\) is the common ratio.

Check the absolute value of the common ratio

For the series to have a sum, the absolute value of the common ratio \(r\) should be less than 1, i.e., \(|r| < 1\). This is the equality condition for a geometric series to converge.

Determine whether the series has a sum

If the absolute value of the common ratio \(r\) is less than 1 (i.e., \(|r| < 1\)), the series converges and therefore, has a sum. If the absolute value of the common ratio \(r\) is greater than or equal to 1, the series diverges, and it does not have a sum.

Key concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the preceding term by a fixed, non-zero number, known as the common ratio. For example, consider the sequence 2, 4, 8, 16, and so forth. This sequence is a geometric series because each term is twice the term before it.

In mathematical terms, a geometric series can be represented as \(\sum_{i=1}^{\infty} a r^{i-1}\), where \(a\) stands for the first term in the series and \(r\) is the common ratio. The formula encapsulates the concept that each term is the product of the first term and the common ratio raised to an exponent that increases with each subsequent term.

Common Ratio

The common ratio in a geometric series is the factor by which each term is multiplied to get the next term. It is essential in determining the behavior of the series, such as whether it converges or diverges. To identify the common ratio \(r\), simply divide a term by the previous term (except for the first term). A positive common ratio means the terms will all have the same sign, while a negative ratio will cause the terms to alternate in sign.

The value of the common ratio has significant implications. If \(|r| < 1\), the terms of the series get progressively smaller, approaching zero. However, if \(|r| \geq 1\), the terms will not approach zero, and if \(r > 1\), the terms will become infinitely large as the series progresses.

Series Convergence

Series convergence is a fundamental concept in mathematics that determines whether the infinite sum of a series reaches a finite value as the number of terms increases. With geometric series, the key factor for convergence is the common ratio. If the absolute value of the common ratio (\( |r| \) is less than 1, the series converges; meaning, the sum of its terms approaches a specific number as more terms are added.

This behavior occurs because the individual terms shrink in magnitude and add less to the total sum. On the contrary, if \( |r| \geq 1 \), the series diverges because the terms do not collectively approach a finite total. Their sum instead grows without bound or oscillates endlessly.

Series Sum

The sum of a geometric series is the cumulative total of all its terms. For a converging geometric series, particularly when \( |r| < 1 \), the sum can be found using the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term, and \( r \) is the common ratio. This elegant equation is derived from the nature of geometric progressions and provides a finite value for the series sum.

However, if the series does not converge (when \( |r| \geq 1 \)), seeking a series sum is incongruous, as the series either grows indefinitely or its terms oscillate without approaching a single finite value. Therefore, the concept of a series sum is only meaningful in the context of convergent geometric series.