Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. \(12+18+24+30+\ldots\)

Step-by-Step Solution

Answer

The series does not have a sum; it's an arithmetic series.

Step 1: Identify the Type of Series

The given series is an arithmetic series because it has a constant difference between consecutive terms. Each term is increasing by 6.

Step 2: Analyze the Series for Convergence

Arithmetic series do not converge. Unlike geometric series, which may converge if they have a common ratio between -1 and 1, arithmetic series grow without bound.

Step 3: Conclusion on Sum Existence

Since the series is arithmetic and the terms increase indefinitely without approaching a specific limit, the series does not have a sum.

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between any two successive terms is constant. This constant difference is known as the "common difference." For example, in the sequence given in the exercise: \(12, 18, 24, 30, \ldots\), the common difference is 6. Each term increases by this constant amount from the previous term.

To identify an arithmetic series, find the difference between any two consecutive terms. If this difference remains consistent throughout the series, then it is arithmetic. This characteristic sets arithmetic series apart from other types of series, like geometric ones.

Understanding the nature of arithmetic series is crucial, particularly in many real-world applications, such as financial calculations and data modeling, where incremental changes are considered. However, keep in mind that arithmetic series have specific properties regarding convergence and sums which are essential to understand.

Convergence

Convergence in series refers to the idea that as you keep summing more terms of the series, the total sum approaches a fixed, finite number. For a series to be convergent, it must not grow infinitely large, and its terms need to approach zero as more terms are added.

Arithmetic series do not typically converge. They have terms that increase (or decrease) linearly with no end, leading to a divergence. In such cases, the series as a whole does not settle on any finite sum. Contrast this with geometric series, where convergence can occur if the common ratio lies between -1 and 1.

• For arithmetic series, since the terms keep growing, the partial sums will also keep growing without stopping.
• Geometric series might converge if the absolute value of the ratio is less than 1.

Understanding the concept of convergence helps in determining whether a series has a sum. Though arithmetic series grow endlessly and do not converge, recognizing this pattern is essential for analyzing series in different contexts.

Sum of a Series

The sum of a series is the aggregate value obtained when you add up all the terms of the series. For some series, particularly finite ones, finding the sum can be straightforward. However, not all series possess a sum, especially those that converge to infinity.

In the case of arithmetic series which are mentioned in the original exercise, since they do not converge, they do not have a finite sum. Every time a term is added, the series grows, further increasing the partial sum without ever approaching a fixed number.

While finite arithmetic series can have a sum calculated using the formula \[ S_n = \frac{n}{2} (a_1 + a_n) \] where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term, this does not apply to infinite series.

• The lack of convergence for the entirety of infinite arithmetic series implies the sum doesn't exist.
• The sum of such series grows infinite, increasing indefinitely as more terms are added.

In summary, while the idea of calculating a sum is integral to understanding series, recognizing when a series, especially an arithmetic one, does not have a finite sum due to non-convergence is equally vital.