Problem 8 - Essential Calculus Early Transcendentals

Question

\(3-8=\) Test the series for convergence or divergence. $$ \sum_{n=1}^{\infty}(-1)^{n} \cos \left(\frac{\pi}{n}\right) $$

Step-by-Step Solution

Answer

The series diverges because the Alternating Series Test fails.

Step 1: Identify Series Type

The series in question is \( \sum_{n=1}^{\infty}(-1)^{n} \cos \left(\frac{\pi}{n}\right) \). Noticing \((-1)^n\), we see this is an alternating series.

Step 2: Apply Alternating Series Test Conditions

The Alternating Series Test states a series \( \sum_{n=1}^{\infty}(-1)^n b_n \) converges if: (1) \( b_n > 0 \) for all \( n \), (2) \( b_{n+1} \leq b_n \) for all \( n \), and (3) \( \, \lim_{{n \to \infty}} \, b_n = 0 \). Here, \( b_n = \cos \left(\frac{\pi}{n}\right) \).

Step 3: Check Positivity Condition

Examining \( \cos \left(\frac{\pi}{n}\right) \): since \( \frac{\pi}{n} \) is in the range \( (0, \pi) \) for \( n \geq 1 \), \( \cos \left(\frac{\pi}{n}\right) \) is positive. Thus, condition (1) is satisfied.

Step 4: Verify Decreasing Sequence Condition

For condition (2), we need to check if \( \cos \left(\frac{\pi}{n+1}\right) \leq \cos \left(\frac{\pi}{n}\right) \). Given \( \cos(x) \) is a decreasing function in \((0, \pi)\), this holds true since \( \frac{\pi}{n+1} > \frac{\pi}{n} \). Thus, condition (2) is satisfied.

Step 5: Evaluate the Limit of Terms

Lastly, we evaluate \( \, \lim_{{n \to \infty}} \, \cos \left(\frac{\pi}{n}\right) \). As \( n \) approaches infinity, \( \frac{\pi}{n} \to 0 \), and \( \cos(0) = 1 \). However, for convergence of an alternating series, this should equal zero, which is not the case here.

Step 6: Conclusion on Convergence

Since condition (3) of the Alternating Series Test is not met (the limit does not equal zero), the series \( \sum_{n=1}^{\infty}(-1)^{n} \cos \left(\frac{\pi}{n}\right) \) diverges.

Key Concepts

Convergence

In mathematical terms, convergence refers to the idea of whether a sequence or series approaches a specific value as its terms progress indefinitely. When we talk about a series converging, we mean that the sum of its infinite terms tends towards a finite number.
Understanding convergence is crucial when working with series because it tells us if our effort of summing infinitely many terms is worthwhile in terms of reaching a meaningful result.

• Limits: The concept of limit is central to determining convergence. We check the limit of the series terms.
• Finite Sum: If the series converges, it sums to a finite value.

When analyzing a series for convergence, we apply specific tests or rules that guide us through the process. These aids can help us determine if a series will sum to a finite value or not.

Divergence

Divergence, on the flip side of convergence, occurs when a sequence or series does not tend to a fixed limit or finite sum.
A divergent series will either grow indefinitely, oscillate, or behave erratically without settling to any constant.

• Behavior Pattern: A divergent series might increase indefinitely or oscillate without delivering a fixed outcome.
• Critical Understanding: Understanding divergence is essential because it tells us that the series does not yield a final, meaningful sum.

In our case, for a series to be divergent, the conditions for convergence must not be fully satisfied.
It’s important for students to recognize convergence and divergence as two outcomes when summing an infinite series.

Series Test

A series test comprises the strategies and methodologies employed to ascertain whether a series converges or diverges.
There are several tests, each with conditions that, when satisfied, provide insight into the nature of the series.

• D’Alembert’s Ratio Test: This is often used for series with factorials or exponential terms.
• Root Test: Helpful when terms involve power functions.
• Alternating Series Test: Specifically for series with alternating positive and negative terms.

Each test serves a particular type of series, allowing for more accurate analysis.
Knowing which test to apply requires understanding the form and structure of the series presented.

Alternating Series Test

The Alternating Series Test helps determine the convergence of series featuring alternating positive and negative terms. It specifically applies when the series takes the form \[ \sum_{n=1}^{\infty}(-1)^n b_n \]where each term alternates in sign.

• Positivity: Check that each term, ignoring the sign, remains positive.
• Decreasing Sequence: Ensure that the terms decrease in absolute value as the series progresses.
• Limit of Zero: Verify that the limit of the absolute terms as \( n \rightarrow \infty \) is zero.

The Alternating Series Test provides a precise manner to investigate series with fluctuating signs, helping determine if it is convergent.
However, if any of these conditions are not satisfied, the series is likely divergent as shown in the original exercise.