Problem 55 - Precalculus: Mathematics for Calculus

Question

Use a graphing calculator to evaluate the sum. \(\sum_{k=1}^{10} k^{2}\)

Step-by-Step Solution

Answer

The sum \( \sum_{k=1}^{10} k^2 = 385 \).

Step 1: Set Up the Series

The series to evaluate is given by the expression \( \sum_{k=1}^{10} k^2 \). This series represents the sum of the squares of the first 10 positive integers.

Step 2: Identify the Function

The function to be evaluated is \( k^2 \) where \( k \) ranges from 1 to 10. We will be calculating the square of each integer within this range.

Step 3: Use Graphing Calculator to Input the Series

On your graphing calculator, enter the sequence of the squared terms: \( 1^2, 2^2, 3^2, ..., 10^2 \). You may use the summation feature, if available, to directly compute \( \sum_{k=1}^{10} k^2 \).

Step 4: Calculate Each Square

Individually calculate each square: for example, \( 1^2 = 1 \), \( 2^2 = 4 \), \( 3^2 = 9 \), ..., and \( 10^2 = 100 \).

Step 5: Sum the Values

Add all the squared values together: 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100.

Step 6: Finalize Result

Compute the total sum of these squared values, obtaining the result of \( 385 \).

Key Concepts

Summing Series

When we talk about "summing a series," it refers to the process of adding a sequence of numbers generated by a specific pattern or formula. In this exercise, the series we are summing has each term as the square of consecutive integers from 1 to 10. The notation \( \sum_{k=1}^{10} k^2 \) indicates that we need to add the squares of all integers starting from 1 up to 10. Here's a simple breakdown of the summing process:

• Identify the series: \( \sum_{k=1}^{10} k^2 \) in this case.
• Calculate each term individually: find \( k^2 \) for each \( k \).
• Add all the terms: \( 1^2 + 2^2 + 3^2 + \, \ldots \, + 10^2 \).

By summing these terms, you obtain a cumulative total, which represents the complete series' value.

Square of Integers

Understanding the "square of integers" is essential for this exercise. Squaring an integer means multiplying it by itself. This operation is visually expressed as \( k^2 \), where \( k \) is an integer. For example:

• \( 1^2 = 1 \times 1 = 1 \)
• \( 2^2 = 2 \times 2 = 4 \)
• \( 3^2 = 3 \times 3 = 9 \)
• \(\vdots\)
• \( 10^2 = 10 \times 10 = 100 \)

Each integer, when squared, provides a part of the entire series that is being summed. Remember, squaring is a componential building block for evaluating mathematical expressions like the series we are discussing.

Evaluation with Technology

Technology, such as a graphing calculator, simplifies the process of evaluating mathematical series. A graphing calculator is an excellent tool for computing complex series with ease. Here's how to make good use of it:

• Input the series expression directly into the calculator, using the summation function if it has one.
• For this exercise, enter \( \sum_{k=1}^{10} k^2 \) into the calculator.
• Let the calculator compute the sum, which saves time and reduces computational mistakes.

Using technology enhances your accuracy and efficiency, making it easier to handle more challenging or lengthy calculations without the risk of manual errors.

Mathematical Series

A "mathematical series" is a sum of terms that follow a particular rule. There are specific types of series such as arithmetic, geometric, and in this case, a power series like the sum of squares. Understanding how to handle series is crucial in mathematics for analyzing patterns and solving real-world problems.
One main feature of a series is its "formula." For the sum of squares, there's a known formula: \[ S_n = \frac{n(n + 1)(2n + 1)}{6} \]where \( n \) is the number of terms. Applying it to \( n = 10 \), you can verify the calculated result directly:

• Plug \( n = 10 \) into the formula.
• Calculate to confirm the result: \( S_{10} = \frac{10(10 + 1)(2 \times 10 + 1)}{6} = 385 \).

This method offers a theoretical check to ensure your manually or technologically derived sum is correct, adding an element of cross-verification to assure solution accuracy.