Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the value of the indicated sum. $$ \sum_{n=1}^{6} n \cos (n \pi) $$

Step-by-Step Solution

Answer

The sum is 3.

Step 1: Understand the Sum Notation

The notation \( \sum_{n=1}^{6} n \cos (n \pi) \) represents the sum of the expression \( n \cos (n \pi) \) as \( n \) goes from 1 to 6. This means we will evaluate \( n \cos (n \pi) \) for each integer value of \( n \) from 1 to 6.

Step 2: Evaluate \( \cos(n\pi) \)

Recognize the pattern of \( \cos(n\pi) \). For even \( n \), \( \cos(n\pi) = 1 \), and for odd \( n \), \( \cos(n\pi) = -1 \). This alternates based on whether \( n \) is odd or even.

Step 3: Calculate Each Term

Calculate each term in the sum:- When \( n = 1 \), \( 1 \cos(1\pi) = -1 \)- When \( n = 2 \), \( 2 \cos(2\pi) = 2 \)- When \( n = 3 \), \( 3 \cos(3\pi) = -3 \)- When \( n = 4 \), \( 4 \cos(4\pi) = 4 \)- When \( n = 5 \), \( 5 \cos(5\pi) = -5 \)- When \( n = 6 \), \( 6 \cos(6\pi) = 6 \)

Step 4: Add the Terms

Add the values from the previous step together to find the sum:\( -1 + 2 - 3 + 4 - 5 + 6 = 3 \).

Key Concepts

Trigonometric Functions in Calculus

Trigonometric functions like sine and cosine play a significant role in calculus. These functions help analyze periodic phenomena and oscillations within different mathematical contexts.

The cosine function, denoted as \( \cos(x) \), has a distinctive periodic behavior, usually expressed as \( \cos(x + 2\pi) = \cos(x) \), showing that it repeats its values every \( 2\pi \).

More specifically, when looking at the expression \( \cos(n\pi) \), a pattern emerges:

• For even \( n \), \( \cos(n\pi) = 1 \)
• For odd \( n \), \( \cos(n\pi) = -1 \)

This alternation is key in problems involving sums with even-odd behavior, leading to alternating series as encountered in our example.

Alternating Series

An alternating series is a series whose terms alternate in sign. This can be incredibly useful when determining convergence in complex sums. Alternating series often include a sequence where terms are added and subtracted in an orderly fashion.

Consider the series given by \( \sum_{n=1}^{6} n \cos(n\pi) \). Here, the alternation stems from the \( \cos(n\pi) \), where even-odd values shift the cosine between 1 and -1. The pattern results in alternating positive and negative multiplication with the values of \( n \).

This kind of series is typical in calculus, especially when using trigonometric functions to express oscillatory behavior, and is essential to mastering the manipulation of series in mathematical applications.

Sum of Series

Finding the sum of a series involves adding a sequence of terms together. Each term reflects a calculated expression based on an underlying pattern or rule.

In our given example, we determined each term by first finding \( n \cos(n\pi) \) for \( n \) from 1 to 6:

• \(1 \cos(1\pi) = -1 \)
• \(2 \cos(2\pi) = 2 \)
• \(3 \cos(3\pi) = -3 \)
• \(4 \cos(4\pi) = 4 \)
• \(5 \cos(5\pi) = -5 \)
• \(6 \cos(6\pi) = 6 \)

Finally, by adding these results: \(-1 + 2 - 3 + 4 - 5 + 6\), we arrive at the sum: 3. This methodical approach ensures no term is overlooked, and the pattern justifies the resulting sum.