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Chapter 5: Problem 3

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

Short Answer

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

Step by step solution

01

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.
02

Substitute Values for k

Each \(k\) value should be substituted into the cosine function. This gives us the terms: \(\cos(\pi), \cos(2\pi), \cos(3\pi), \text{ and } \cos(4\pi)\).
03

Evaluate Each Cosine

\[\begin{align*}\cos(\pi) &= -1, \\cos(2\pi) &= 1, \\cos(3\pi) &= -1, \\cos(4\pi) &= 1.\end{align*}\]
04

Add the Terms

Add the results of the cosine evaluations: \(-1 + 1 + (-1) + 1 = 0\).
05

Write the Final Sum

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In Exercises \(91-94\) , you will find the area between curves in the plane when you cannot find their points of intersection using simple algebra. Use a CAS to perform the following steps: a. Plot the curves together to see what they look like and how many points of intersection they have. b. Use the numerical equation solver in your CAS to find all the points of intersection. c. Integrate \(|f(x)-g(x)|\) over consecutive pairs of intersection values. d. Sum together the integrals found in part (c). $$ f(x)=\frac{x^{3}}{3}-\frac{x^{2}}{2}-2 x+\frac{1}{3}, \quad g(x)=x-1 $$

Find the areas of the regions enclosed by the lines and curves in Exercises \(51-58 .\) $$ x-y^{2 / 3}=0 \quad \text { and } \quad x+y^{4}=2 $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \frac{6 \cos t}{(2+\sin t)^{3}} d t $$

Find the areas of the regions enclosed by the lines and curves in Exercises \(41-50 .\) $$ y=2 x-x^{2} \quad \text { and } \quad y=-3 $$

In Exercises \(89-92,\) use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100,200,\) and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$ f(x)=x \sin \frac{1}{x} \quad \text { on } \quad\left[\frac{\pi}{4}, \pi\right] $$
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