Problem 100 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to find the sum. $$\sum_{j=1}^{10} \frac{6}{3 j+1}$$

Step-by-Step Solution

Answer

The sum of the series can only be specifically determined using a graphing utility to compute, since the provided exercise emphasizes the usage of such. Therefore the solution will depend on the result from the graphing utility and isn't provided here.

Step 1: Understand the Sigma Notation

The given expression is in summation form, also known as sigma notation. The symbol \(\sum\), a Greek capital letter sigma, represents the sum. The number at the bottom of the sigma, in this case 'j=1', is the initial index. The number above the sigma, '10' in this case, is the final index. For each value of 'j' from 1 to 10, the expression to the right of the sigma \(\frac{6}{3j+1}\), will be evaluated, and these results will be added together to get the total sum.

Step 2: Enter the Series into the Graphing Utility

Now, input the series \(\frac{6}{3j+1}\) from 'j=1' to 'j=10' into a graphing utility. This may look something like 'sum(6/(3j+1), j, 1, 10)' depending on the specific calculator or utility being used.

Step 3: Calculate the Sum

Once the series is entered correctly, use the utility to calculate the sum. The utility should provide a numerical result representing the sum of the sequence from 'j=1' to 'j=10.'

Key Concepts

Sigma Notation

Sigma notation is a concise and efficient way to represent the sum of a series of terms. It is symbolized by the Greek capital letter sigma (\textsigma). When faced with a series like \(\sum_{j=1}^{10} \frac{6}{3j+1}\), understanding each component is vital.

In this example, the 'j=1' beneath the sigma indicates the starting index, while the '10' above denotes the end index. This tells us that the series runs from 'j=1' to 'j=10'. For each integer 'j' within this range, we evaluate the expression \(\frac{6}{3j+1}\) and sum all those evaluations. The numerator '6' is a constant, while the denominator '3j+1' changes as 'j' increments from 1 to 10.

For students working with sigma notation, it's beneficial to write out the first few terms manually to get a feel for the pattern before using a graphing utility or calculator. This aids in comprehending the rate at which the series grows and the impact of each term on the sum.

Graphing Calculators

Graphing calculators are incredibly useful tools for students dealing with complex mathematics, particularly when it comes to series and sequences. To calculate the sum of a series like \(\sum_{j=1}^{10} \frac{6}{3j+1}\), a graphing calculator can significantly streamline the process.

Entering the Series

First, the series must be input correctly. The syntax depending on the graphing calculator model you have could be something like `sum(6/(3*j+1), j, 1, 10)`. It's crucial to understand your specific calculator's syntax to avoid errors in computation.

Using the Sum Function

After inputting the series, the calculator's sum function handles the tedium of evaluating each term and accumulating the total sum. It's important to verify if the calculator is set to the right mode (e.g., radians or degrees) if any trigonometric functions are involved in the series to ensure accurate results.

Interpreting the Results

Upon obtaining the result, interpret it in context. In some cases, you may need to round off to a certain number of decimal places or continue with the evaluation as part of a larger problem. Using graphing calculators not only saves time but helps in visual learning by plotting sequences when needed.

Series and Sequences

In mathematics, series and sequences are fundamental concepts often discussed together. A sequence is an ordered list of numbers following a specific rule, while a series is the sum of the terms of a sequence.

For example, the sequence defined by the general term \(\frac{6}{3j+1}\) would list the numbers generated by substituting 'j' with integers from 1 to 10. When these terms are added, they form a series. In our exercise, this summation is condensed into sigma notation, which we calculate to be a finite sum since it has a starting and an endpoint.

Understanding the behavior of sequences, such as whether they converge or diverge as they progress, can provide insight into the sum of the series. Especially in engineering and sciences, knowing how to sum sequences using formulas, sigma notation, and graphing calculators is critical for processing data and solving real-life problems.