Problem 103 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$\frac{1}{3(1)}+\frac{1}{3(2)}+\frac{1}{3(3)}+\cdots+\frac{1}{3(9)}$$

Step-by-Step Solution

Answer

The sum written in sigma notation is \(\frac{1}{3} \sum_{n=1}^{9} \frac{1}{n}\). The exact sum value could be computed using a graphing utility and would be around 1.67.

Step 1: Understand the sequence

From the given sum, we see that this is a sequence where the denominator starts from 1, and increases by 1 in steps. We can represent this sequence as \(1/n\), where \(n\) ranges from 1 to 9.

Step 2: Write the sum in sigma notation

Given the sequence is \(1/n\), the sum can be written in sigma notation as \(\frac{1}{3} \sum_{n=1}^{9} \frac{1}{n}\). Sigma notation allows to express long sums in a concise manner. Here, \(\sum_{n=1}^{9}\) indicates that we are summing the sequence over \(n\) from 1 to 9.

Step 3: Use a graphing utility to compute the sum

The next step involves using a graphing utility to compute the sum. Because the graphing utility might differ from individual to individual, general steps are: input the sum as written into the calculator, then press the 'compute' or 'calculate' button, or equivalent on device, to obtain the sum.

Key Concepts

Sequence and Series

In mathematics, sequences and series form a foundational concept that helps us understand patterns in numbers. A **sequence** is a list of numbers arranged in a specific order. For example, in this exercise, the sequence is given by \( \frac{1}{3(1)}, \frac{1}{3(2)}, \frac{1}{3(3)}, \ldots, \frac{1}{3(9)} \). Here, each term in the sequence is expressed as \( \frac{1}{3n} \), where \( n \) progressively increases from 1 to 9.
As you explore these sequences, imagine them as stepping stones, where each stone—or term—comes one after another following a set rule.
An extension of sequences leads us to a **series**, which is essentially the sum of the terms of a sequence."

Sum of Series

The process of finding the sum of a series is crucial in mathematics, particularly when handling large data sets or long sequences. In the context of the exercise, we use a concept called **sigma notation** to represent this sum compactly and effectively.
Sigma notation uses the Greek letter \( \Sigma \), encapsulating the entire sum. For instance, the given sequence \( \frac{1}{3(1)} + \frac{1}{3(2)} + \ldots + \frac{1}{3(9)} \) can be written as \( \frac{1}{3} \sum_{n=1}^{9} \frac{1}{n} \).
This notation allows mathematicians to express long sums in a leaner and more manageable format.

• \( \Sigma \): Represents the sum over a series.
• The expression next to \( \Sigma \): Indicates the terms of the sequence to be summed.
• The limits below and above \( \Sigma \): Show where the sum begins and ends.

Understanding this method simplifies the addition process, especially when the terms follow a clear pattern.

Graphing Utility

Graphing utilities serve as a technological aid in solving mathematical problems, especially when dealing with large sequences or complex functions. These tools can be found in scientific calculators, specialized computer software, or online platforms. In tackling series sums, a graphing utility can perform calculations swiftly, providing precise results with minimal error.
To use a graphing utility:

• First, input the sigma notation as specified in the exercise, \( \frac{1}{3} \sum_{n=1}^{9} \frac{1}{n} \).
• Next, find the 'compute' or 'calculate' feature on your device to process the input.
• Ensure to follow any device-specific instructions for entering functions and operations.

While practices might vary depending on the device, the core function remains the same: delivering accurate results to facilitate complex calculations. Leveraging these tools allows for a deeper exploration of mathematical concepts without getting bogged down by extensive manual calculations.