Problem 10 - Essential Calculus

Question

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{1}{n} \cos n \pi $$

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{1}{n} \cos n\pi \) converges.

Step 1: Identify the type of the series

Establish that this is an alternating series. An alternating series is defined as one that alternates between positive and negative terms. This is true for \( \sum_{n=1}^{\infty} \frac{1}{n} \cos n\pi \) because \(\cos n\pi\) alternates between 1 and -1 as n varies between even and odd numbers.

Step 2: Applying the Alternating Series Test

The Alternating Series Test states that the alternating series \(\sum_{n=1}^{\infty} (-1)^n a_n\), where each \(a_n > 0\), converges if it satisfies two conditions: (1) The terms \(a_n\) are decreasing and (2) The limit as n approaches infinity of \(a_n\) equals zero. Here \(a_n = \frac{1}{n}\). The terms \(a_n = \frac{1}{n}\) are clearly decreasing and its limit as n approaches infinity is indeed zero.

Step 3: Conclude

Based on the Alternating Series Test, the series \( \sum_{n=1}^{\infty} \frac{1}{n} \cos n\pi \) converges since both conditions of the test are satisfied. It is important to remember this conclusion relies on the series being an alternating series and meeting all the conditions of the Alternating Series Test.

Key Concepts

Alternating Series

A series is called an alternating series if its terms alternate in sign. This means that successive terms flip between positive and negative. Alternating series are common in mathematics and are often represented in the form \[\sum_{n=1}^{\infty} (-1)^n a_n\]where the terms \(a_n\) are all positive.
For example, in the series \[\sum_{n=1}^{\infty} \frac{1}{n} \cos n\pi\]\(\cos n\pi\) alternates between 1 and -1 whenever \(n\) is an even or odd number. This makes the series an alternating series, as the positive-negative pattern ensures there's a regular flip in signs.
Understanding alternating series is key for determining their overall behavior, which presents us with a great tool to analyze convergence.

Alternating Series Test

The Alternating Series Test is an essential tool for determining the convergence of alternating series. It helps us verify if a given series converges based on specified criteria.
For a series to pass the Alternating Series Test and thus converge, it must meet two conditions:

• The absolute value of the terms \(a_n\) must be decreasing as \(n\) increases. In simpler words, each term should be smaller than the one before it.
• The limit of the terms \(a_n\) as \(n\) approaches infinity should be zero.

In our example: \[\sum_{n=1}^{\infty} \frac{1}{n} \cos n\pi\]the terms decrease because \(a_n = \frac{1}{n}\) decreases as \(n\) gets larger. Plus, \(\lim{{n \to \infty}} \frac{1}{n} = 0\). Thus, it's confirmed that this particular series converges by the Alternating Series Test.
Using the Alternating Series Test helps us confidently establish convergence for series that initially appear complex.

Series Divergence

Unlike convergent series, divergent series do not approach a finite value as their number of terms increases towards infinity. Divergence can occur for several reasons and identifying it is crucial to understanding the behavior of a series.
For series that we think might be divergent, tests such as the Alternating Series Test can highlight conditions leading to divergence. If a series does not meet these conditions, it can be divergent.
While our example's alternating series does converge because it passed all required conditions, recognizing the potential for divergence—when conditions like decreasing terms and zero limits are unmet—is essential. This prepares us for broader analysis involving series without straightforward outcomes.