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Chapter 5: Problem 12

Write out each term of the summation and compute the sum. $$ \sum_{i=0}^{5}(-1)^{i} \cos (\pi i) $$

Short Answer

Expert verified
The sum is 6.

Step by step solution

01

Understand the Summation

The summation notation \( \sum_{i=0}^{5} (-1)^{i} \cos(\pi i) \) expands terms using \( i \) starting from 0 up to 5. We have to compute each term separately.
02

Evaluate Terms Individually

To calculate each term of the summation, substitute \( i \) from 0 to 5 into \( (-1)^{i} \cos(\pi i) \):- When \( i = 0 \): \((-1)^0 \cos(\pi \times 0) = 1 \times 1 = 1\)- When \( i = 1 \): \((-1)^1 \cos(\pi \times 1) = -1 \times (-1) = 1\)- When \( i = 2 \): \((-1)^2 \cos(\pi \times 2) = 1 \times 1 = 1\)- When \( i = 3 \): \((-1)^3 \cos(\pi \times 3) = -1 \times (-1) = 1\)- When \( i = 4 \): \((-1)^4 \cos(\pi \times 4) = 1 \times 1 = 1\)- When \( i = 5 \): \((-1)^5 \cos(\pi \times 5) = -1 \times (-1) = 1\)
03

Sum the Evaluated Terms

Add up all the computed terms:\(1 + 1 + 1 + 1 + 1 + 1 = 6\). Therefore, the sum of the series is 6.

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Key Concepts

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

Trigonometric Functions
Trigonometric functions are a key part of mathematics, often used in geometry and analysis. They relate the angles of a triangle to the lengths of its sides in the field of trigonometry. Here are some of the most important aspects:
  • Sine ( ext{sin}) and cosine ( ext{cos}) functions are the most fundamental trigonometric functions. They are periodic, meaning they repeat their values in regular intervals.
  • Angles can be measured in degrees or radians, with radians often being used in calculus for more straightforward calculations.
  • The cosine function, which we are focusing on, is defined for an angle \( \theta \) as the adjacent side divided by the hypotenuse in a right-angled triangle.
The cosine function has a range from -1 to 1 and is even, meaning \( ext{cos}(-\theta) = ext{cos}(\theta) \).
Summation Notation
Summation notation, represented by the Greek letter sigma (\( \sum \)), is a convenient way to express long sums. Instead of writing every single term, you can use this notation to imply repeated addition.
  • The expression \( ext{\( \sum_{i=0}^{5} a_i \)} \) means summing up all \( a_i \) values from \( i = 0 \) to \( i = 5 \).
  • Each term in the summation is found by substituting different values of \( i \) into the expression inside the sum.
  • In the context of the previously given exercise, the summation runs from \( i=0 \) to \( i=5 \) for \( (-1)^{i} ext{cos}(\pi i) \).
This compact method makes handling large sequences manageable and easier to analyze.
Cosine Function
The cosine function, written as \( ext{cos} \), is a periodic function that oscillates between -1 and 1. It is noteworthy for several reasons:
  • The cosine of an angle \( ext{\( n\pi \)} \) is 1 if \( n \) is even and -1 if \( n \) is odd. This is based on the unit circle perspective.
  • In our summation exercise, the cosine function uses the property \( ext{cos}( \pi i) \), which simplifies based on whether \( i \) is even or odd because \( ext{cos}(n\pi) = (-1)^n\).
  • This periodic behavior helps simplify computations in series like the one given, where the multiplier \( (-1)^i \) affects only the sign and not the outcome magnitude.
Understanding these properties can greatly simplify working with functions in trigonometric problems.

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Most popular questions from this chapter

A definite integral is given. (a) Approximate the definite integral with the Trapezoidal Rule and \(n=4\) (b) Approximate the definite integral with Simpson's Rule and \(n=4\) (c) Find the exact value of the integral. $$ \int_{0}^{3}\left(x^{3}+2 x^{2}-5 x+7\right) d x $$

Approximate the definite integral with the Trapezoidal Rule and Simpson's Rule, with \(n=6\). $$ \int_{-1}^{1} e^{x^{2}} d x $$

Find \(n\) such that the error in approximating the given definite integral is less than 0.0001 when using: (a) the Trapezoidal Rule (b) Simpson's Rule $$ \int_{0}^{5} x^{4} d x $$

Approximate the definite integral with the Trapezoidal Rule and Simpson's Rule, with \(n=6\). $$ \int_{1}^{4} \ln x d x $$

Evaluate the definite integral. $$ \int_{\pi / 2}^{\pi} \cos x d x $$
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