Problem 50 - Precalculus: Mathematical for Calculus

Question

Use a graphing calculator to evaluate the sum. $$\sum_{j=5}^{15} \frac{1}{j^{2}+1}$$

Step-by-Step Solution

Answer

Use the calculator's summation function to find the sum.

Step 1: Identify the Series

The exercise requires evaluating the sum \( \sum_{j=5}^{15} \frac{1}{j^2+1} \). This is a finite series starting from \( j=5 \) to \( j=15 \), where the general term is \( \frac{1}{j^2+1} \).

Step 2: Setup the Graphing Calculator

Turn on your graphing calculator and access the summation function, often found under the math or calculus menu, denoted by a symbol resembling \( \Sigma \).

Step 3: Input the Series

In the summation function, set the lower limit to \( 5 \) and the upper limit to \( 15 \). Enter the expression \( \frac{1}{j^2+1} \) as the function to be summed, ensuring that the variable is set to \( j \).

Step 4: Calculate the Sum

After inputting the correct series parameters, press the calculate or enter button on your graphing calculator to compute the sum from \( j=5 \) to \( j=15 \).

Step 5: Interpret the Result

Review the result displayed on the graphing calculator. This is the evaluated sum of the series \( \sum_{j=5}^{15} \frac{1}{j^2+1} \). Ensure the value is reasonable given the terms involved.

Key Concepts

Using a Graphing Calculator

A graphing calculator is a powerful tool for evaluating mathematical expressions, especially when dealing with finite series. It allows you to visualize mathematical concepts and compute values that might be complex or time-consuming if done manually. To use a graphing calculator effectively, you should first familiarize yourself with its various features and menus. Most graphing calculators have a dedicated function for summation, often symbolized by \( \Sigma \), and you can find it in the math or calculus sections of the menu.

• Turn on your graphing calculator and navigate to the summation feature.
• Input the correct parameters such as the lower and upper limits for the series.
• Ensure the variable matches your series notation before calculating.

Once your inputs are checked, you can proceed to compute the sum, obtaining an accurate result efficiently. This tool is not only useful for evaluating expressions but also for learning and visualizing mathematical concepts in real-time.

Understanding Summation Notation

Summation notation, indicated by the sigma symbol \( \Sigma \), represents the sum of a sequence of terms. It is a compact way to express long sums, allowing one to see the range of the series and the explicit general term involved.

• \( j=5 \) is the starting index, and \( j=15 \) is the ending index.
• The general term \( \frac{1}{j^2+1} \) defines the expression to be summed for each \( j \) value.
• The expression within the summation indicates a functional relationship or rule that generates each term.

Summation notation is particularly useful for providing clarity and conciseness in mathematical communication, making it easy to recognize patterns and limits within sequences, leading to a better understanding of series.

How to Evaluate the Expression

Evaluating a finite series involves calculating the sum of its terms based on given limits and a general term. In the exercise, the expression \( \frac{1}{j^2+1} \) is summed from \( j=5 \) to \( j=15 \). To evaluate this:

• Identify the series' limits, which are \( j=5 \) to \( j=15 \).
• Substitute each integer value within the limits into the given expression.
• Sum these computed values sequentially.

By using a graphing calculator, this process becomes significantly easier, as you input the entire series setup, and the calculator performs the computations, displaying the sum. Evaluating the expression manually or with a calculator deepens comprehension of how series work and how functions are evaluated over specific intervals.