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 Cosine Function

The function \( \cos(n \pi)\) involves calculating the cosine of multiples of \( \pi\). Recall that \( \cos(n \pi)\) alternates between -1 and 1 based on whether \(n\) is even or odd. Specifically, \( \cos(n \pi) = (-1)^n\).

Step 2: Calculate Terms

For each \(n = 1, 2, 3, 4, 5, 6\), calculate the term \(n \cos(n \pi)\): - \(n = 1\), \(1 \times \cos(1 \pi) = 1 \times (-1)^1 = -1\)- \(n = 2\), \(2 \times \cos(2 \pi) = 2 \times (-1)^2 = 2\)- \(n = 3\), \(3 \times \cos(3 \pi) = 3 \times (-1)^3 = -3\)- \(n = 4\), \(4 \times \cos(4 \pi) = 4 \times (-1)^4 = 4\)- \(n = 5\), \(5 \times \cos(5 \pi) = 5 \times (-1)^5 = -5\)- \(n = 6\), \(6 \times \cos(6 \pi) = 6 \times (-1)^6 = 6\).

Step 3: Sum the Calculated Terms

Sum all calculated terms together: \[-1 + 2 - 3 + 4 - 5 + 6\]. Calculate the sum step-by-step:- \(-1 + 2 = 1\)- \(1 - 3 = -2\)- \(-2 + 4 = 2\)- \(2 - 5 = -3\)- \(-3 + 6 = 3\) The total sum is \(3\).

Key Concepts

Cosine Function

In trigonometry, the cosine function is a fundamental concept that relates the angle of a triangle to the ratio of the adjacent side to the hypotenuse. When dealing with angles that are multiples of \( \pi \), such as in the given exercise, it becomes essential to understand how cosine behaves. The key takeaway is that for any integer \( n \), \( \cos(n\pi) \) will alternate between -1 and 1.

The cosine values alternate because of the periodic nature of the cosine function, specifically as it relates to even and odd multiples of \( \pi \). To break it down:

• When \( n \) is odd, \( \cos(n\pi) = -1 \).
• When \( n \) is even, \( \cos(n\pi) = 1 \).

This alternating property is very useful in various mathematical applications, including series and periodic functions. It's mainly because the unit circle where cosine is derived shows symmetry along these specific angles.

Alternating Series

An alternating series is a series where the terms change signs alternately. The alternating sign in the series adds an interesting layer of complexity, which can impact the sum's convergence and calculation. For our example in the exercise:

The series formed by \( \sum_{n=1}^{6} n \cos(n \pi) \) is alternating, since the cosine function introduces alternating negative and positive terms. This means terms switch between positive and negative, like:

• \( 1 \times \cos(1\pi) = -1 \)
• \( 2 \times \cos(2\pi) = 2 \)
• \( 3 \times \cos(3\pi) = -3 \)
• \( \ldots \)

The alternating nature of the terms is determined by the parity of \( n \) (whether \( n \) is odd or even), affecting the sign of each term according to the rule we derived for \( \cos(n\pi) \).

Understanding alternating series is crucial when dealing with sums that might not simply add or subtract without producing surprises in their outcomes. Alternating series also feature prominently in calculus, especially in the context of convergence.

Summation Notation

Summation notation provides a compact way to represent the sum of a sequence of numbers and is critically used across all branches of mathematics. In our exercise, we see the notation \( \sum_{n=1}^{6} \) which signals the addition of terms from \( n = 1 \) to \( n = 6 \).

This notation is important for several reasons:

• It reduces complexity in problems by avoiding the need to write long expressions repeatedly.
• Allows mathematicians to clearly express the operation they're carrying out, particularly in sequences and series.

The bottom number of the summation symbol \( \sum \) indicates the starting value for \( n \), while the top number shows the end value.

The structure, \( n \cos(n\pi) \), inside the summation symbol tells us what to calculate at each step. This clear structure simplifies understanding and solving problems that involve sequences or series. Summation notation makes the process of solving complex trigonometric sums more systematic and less cumbersome.