Problem 20 - Thomas Calculus

Question

Which of the series in Exercises \(17-56\) converge, and which diverge? Use any method, and give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{1+\cos n}{n^{2}} $$

Step-by-Step Solution

Answer

The series converges by the Comparison Test.

Step 1: Identify the Series Type

The given series is \( \sum_{n=1}^{\infty} \frac{1+\cos n}{n^2} \). This is an infinite series that can be associated with the series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), which is a p-series.

Step 2: Analyze Using Comparison Test

Recall the comparison test: if \( \sum a_n \) is a series of positive terms, and \( 0 \leq a_n \leq b_n \) for all \( n \) from some point onward, and if \( \sum b_n \) converges, then \( \sum a_n \) also converges.In our case, use \( a_n = \frac{1 + \cos n}{n^2} \) and \( b_n = \frac{2}{n^2} \), noting \( 0 \leq \cos n \leq 1 \) means \( 1 + \cos n \leq 2 \). The series \( \sum \frac{2}{n^2} \) is a p-series with \( p = 2 \), which is known to converge.

Step 3: Convergence Conclusion

Since \( 0 \leq \frac{1+\cos n}{n^2} \leq \frac{2}{n^2} \) and \( \sum \frac{2}{n^2} \) is a convergent p-series, by the Comparison Test, the series \( \sum \frac{1+\cos n}{n^2} \) also converges.

Key Concepts

Comparison Test

The comparison test is a useful technique for determining the convergence or divergence of an infinite series. This test is particularly beneficial when you suspect that a given series resembles another series with known convergence properties. It operates on the principle that if a series is squeezed between two other series, its behavior aligns with those series.

Here's how the test works:

• Identify two series: the one you want to analyze, denoted as \( \sum a_n \), and another reference series \( \sum b_n \) with known convergence or divergence.
• For all terms from some point onward, ensure that \( 0 \leq a_n \leq b_n \).
• If \( \sum b_n \) converges, \( \sum a_n \) also converges. Conversely, if \( \sum a_n \) diverges when \( \sum b_n \) diverges.

In the exercise, we used \( a_n = \frac{1 + \cos n}{n^2} \) and compared it to \( b_n = \frac{2}{n^2} \), where \( \cos n \) fluctuates between \(-1\) and \(1\), but for our series, we only need the range \(0\) to \(1\) since we're dealing with positive terms. Thus, establishing \( a_n \leq b_n \) allowed us to deduce convergence through the comparison test.

P-Series

A p-series is a type of mathematical series that takes the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a positive constant. It is an essential concept in understanding convergence of series in calculus.

For p-series:

• Convergence occurs if \( p > 1 \).
• Divergence occurs if \( p \leq 1 \).

In this exercise, the series \( \sum \frac{2}{n^2} \) is identified as a p-series with \( p = 2\). Since \( p = 2 > 1\), it converges. Recognizing this similarity allowed us to apply the comparison test confidently, knowing that the behavior of p-series is well-established.

Trigonometric Series

Trigonometric series incorporate trigonometric functions, such as sine and cosine, in their terms. They often exhibit interesting properties due to the periodic nature of these functions.

In the context of our exercise, we are dealing with \( 1 + \cos n \), where the cosine function's periodic behavior plays a crucial role in how we understand and modify the series. Specifically:

• \( \cos n \) oscillates between \(-1\) and \(1\), but within the series' terms \( 1+\cos n \), it results in values between \(0\) and \(2\).
• This variability influences how we compare the given series with a reference series.

For convergence analysis, it’s key to understand that the oscillating nature of \( \cos n \) does not affect the limit or the bound comparison needed in our particular comparison test.

Infinite Series

Infinite series are sequences of numbers that are added up forever, requiring special techniques to determine whether they settle to a finite sum or not. Many series that appear divergent at first glance might surprisingly converge, and vice versa.

Key features of infinite series:

• Convergence: If the sequence of partial sums \( S_n = a_1 + a_2 + ... + a_n \) approaches a finite limit as \( n \rightarrow \infty \).
• Divergence: If \( S_n \) does not approach a finite limit.

In this exercise, \( \sum_{n=1}^{\infty} \frac{1 + \cos n}{n^2} \) is an example of an infinite series. Using techniques like the comparison test is fundamental in concluding the behavior of such a series. Infinite series can sometimes behave counterintuitively, hence the need for rigorous methods to determine their true nature.