Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the value of the indicated sum. $$ \sum_{k=-1}^{6} k \sin (k \pi / 2) $$

Step-by-Step Solution

Answer

The value of the sum is 4.

Step 1: Understand the Sum Notation

The sum \( \sum_{k=-1}^{6} k \sin (k \pi / 2) \) represents adding the value of the expression \( k \sin (k \pi / 2) \) for each integer \( k \) from \(-1\) to \(6\), inclusive. So, we'll compute \( k \sin (k \pi / 2) \) for each \( k \) in this range and add the results.

Step 2: Calculate Each Term

Let's compute each term individually:- For \( k = -1 \), \( k \sin(- \pi / 2) = -1 \times (-1) = 1 \).- For \( k = 0 \), \( k \sin(0) = 0 \times 0 = 0 \).- For \( k = 1 \), \( k \sin(\pi / 2) = 1 \times 1 = 1 \).- For \( k = 2 \), \( k \sin(\pi) = 2 \times 0 = 0 \).- For \( k = 3 \), \( k \sin(3 \pi / 2) = 3 \times (-1) = -3 \).- For \( k = 4 \), \( k \sin(2\pi) = 4 \times 0 = 0 \).- For \( k = 5 \), \( k \sin(5 \pi / 2) = 5 \times 1 = 5 \).- For \( k = 6 \), \( k \sin(3 \pi) = 6 \times 0 = 0 \).

Step 3: Sum the Terms

Now we sum the computed values: \[1 + 0 + 1 + 0 - 3 + 0 + 5 + 0 = 4\]. Therefore, the value of the sum is 4.

Key Concepts

Sum Notation

The concept of sum notation is pivotal when dealing with a sequence of numbers or expressions that need to be added together. In mathematics, the notation \( \sum \) (sigma) is used to represent this summation process. The initial expression given in the exercise is \( \sum_{k=-1}^{6} k \sin (k\pi / 2) \). This means that we will evaluate the term \( k \sin (k\pi / 2) \) for every integer value of \( k \) starting from -1 and ending at 6.
The indices at the bottom and top of the sigma, here \(-1\) and \(6\), define the range of values that \( k \) can take. Each time we compute the term for a specific \( k \), we then add it to the sum of previous results. This process continues until we've incorporated the value for \( k=6 \) into the sum.
Understanding sum notation not only helps to track what needs to be added, but also forms the foundation for evaluating more complex sequences and series in mathematics. Always begin by identifying the starting and ending indices, and proceed through each term sequentially, ensuring none are skipped.

Trigonometric Identities

Trigonometric identities are mathematical equations that are true for all angles and form the basis for evaluating trigonometric functions like sine, cosine, and tangent. In this exercise, we specifically use the sine function, denoted as \( \sin \). It helps to recognize the periodicity of the sine function, which means the function repeats its values in a regular pattern.
For any angle \( \theta \), \( \sin(\theta) \) has known values at specific points:

• \( \sin(0) = 0 \)
• \( \sin(\pi/2) = 1 \)
• \( \sin(\pi) = 0 \)
• \( \sin(3\pi/2) = -1 \)
• \( \sin(2\pi) = 0 \)

This periodic nature is key when calculating terms for \( k \sin(k\pi/2) \) as \( k \) ranges over different integer values. By recognizing these fixed results, you'll more quickly deduce the outcome of each computation. Knowing these standard sine values simplifies evaluating the function at natural intervals, ensuring you correctly apply the trigonometric identity at each step.

Evaluation of Trigonometric Functions

Evaluating trigonometric functions involves calculating the value of the function for a given angle. In this problem, each \( k \sin(k\pi/2) \) represents such an evaluation of the sine function. The approach consists of multiplying the sine value of the angle \( k\pi/2 \) by \( k \), which depends on the result of the sine function at that angle.
To compute this effectively:

• Substitute each integer \( k \) into the expression \( k\pi/2 \).
• Use known values of the sine function, ensuring computations are efficient and free of miscalculation.
• Multiply the result by \( k \) for each specific \( k \).

For instance, for \( k=3 \), we find \( \sin(3\pi/2) = -1 \), and hence, the term becomes \( 3 \times -1 = -3 \).
Ultimately, accurately evaluating each trigonometric expression is paramount to successfully determining the entire sum. This process relies heavily on both recognizing sine's periodic pattern and sticking closely to ordered calculations for each successive \( k \). Thoroughly following these principles and correctly applying the relevant trigonometric identities ensures precise results.