Problem 3 - Thomas Calculus

Question

Write the sums in Exercises \(1-6\) without sigma notation. Then evaluate them. $$ \sum_{k=1}^{4} \cos k \pi $$

Step-by-Step Solution

Answer

The sum \( \sum_{k=1}^{4} \cos k \pi \) is 0.

Step 1: Understand Sigma Notation

The given expression \( \sum_{k=1}^{4} \cos k \pi \) indicates that we need to evaluate the sum \( \cos(1 \cdot \pi) + \cos(2 \cdot \pi) + \cos(3 \cdot \pi) + \cos(4 \cdot \pi) \). Each term in the series is calculated by substituting \(k\) with integers from 1 to 4 inclusive.

Step 2: Substitute Values for k

Each \(k\) value should be substituted into the cosine function. This gives us the terms: \(\cos(\pi), \cos(2\pi), \cos(3\pi), \text{ and } \cos(4\pi)\).

Step 3: Evaluate Each Cosine

\[\begin{align*}\cos(\pi) &= -1, \\cos(2\pi) &= 1, \\cos(3\pi) &= -1, \\cos(4\pi) &= 1.\end{align*}\]

Step 4: Add the Terms

Add the results of the cosine evaluations: \(-1 + 1 + (-1) + 1 = 0\).

Step 5: Write the Final Sum

The sum of the series evaluated is 0. Thus, \( \sum_{k=1}^{4} \cos k \pi = 0 \).

Key Concepts

Sigma Notation

Sigma notation, often represented by the symbol \( \sum \), provides a concise way to depict a series or sum of terms. It tells us how to sum over several values. In our exercise, the sigma notation \( \sum_{k=1}^{4} \cos k \pi \) means we're summing the cosine function for integer values of \( k \) from 1 to 4.

• The \( k=1 \) at the bottom of the sigma symbol tells us where to start summing.
• The number \( 4 \) at the top indicates where to stop.
• The expression \( \cos k \pi \) is the function being summed, with \( k \) taking each integer value from the starting point to the stopping point.

This notation is highly efficient, especially for large series, as it eliminates the need to write out each term explicitly. In simple terms, sigma notation is a mathematical shorthand to avoid repetitive addition.

Cosine Function

The cosine function is a fundamental element in trigonometry. It is periodic and expresses the horizontal coordinate of a point on a unit circle.

• The function \( \cos \theta \) describes the cosine of an angle \( \theta \), often measured in radians.
• On the unit circle, \( \cos 0 = 1 \) and \( \cos \pi = -1 \).
• Due to its periodic nature, \( \cos(\theta + 2\pi) = \cos \theta \). This property extends further to \( \cos(2\pi) = 1 \) and \( \cos(3\pi) = -1 \).

Understanding the cyclic pattern of cosine values aids in evaluating each term in the series. Every \( \pi \)-shift along the horizontal axis simply alternates the cosine value between -1 and 1.

Series Evaluation

To evaluate a series, break it into manageable parts by calculating each term separately before adding them together. Here, we start with the series from the exercise: 1. List each term produced by substituting \( k \) values into \( \cos k \pi \): - \( \cos(\pi) = -1 \), - \( \cos(2\pi) = 1 \), - \( \cos(3\pi) = -1 \), - \( \cos(4\pi) = 1 \). Next, sum them up step by step:

• \(-1 + 1 + (-1) + 1 \)
• Start by pairing the terms: \(-1 + 1 = 0\), and \(-1 + 1 = 0\)
• Add the results: \(0 + 0 = 0\)

The resulting sum comes out to 0. This evaluation demonstrates the balance in the periodic oscillation of the cosine values.

Mathematical Notation

In mathematics, clear and consistent notation is key to understanding complex ideas easily. Mathematical notation serves as a universal language among mathematicians and students.

• Simplifies the representation of problems, like using sigma notation for sums.
• Symbolic expressions, such as \( \cos \), instantly convey information about trigonometric properties.
• The use of Greek letters, merely such as \( \pi \), links concepts back to geometry and helps in relating angles to circular measurements.

By mastering mathematical notation, students can unlock a more seamless approach to interpreting and solving equations across various topics in math.