Problem 40 - Essential Calculus

Question

Determine whether the series converges conditionally or absolutely, or diverges. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \arctan $$

Step-by-Step Solution

Answer

The given series \(\sum_{n=1}^{\infty}(-1)^{n+1} \arctan(n)\) is conditionally convergent, but not absolutely convergent.

Step 1: Identify the sequence

First step is to identify the sequence for which the series is formed. Here the sequence is \((-1)^{n+1} \arctan(n)\).

Step 2: Apply the Alternating Series Test

The Alternating Series Test states that a series \(\sum (-1)^na_n\), with \(a_n>0\), converges if, and only if, the sequence \({a_n}\) is decreasing and \(\lim_{n \to \infty}a_n = 0\). So, we need to ensure these conditions hold in the given series. \nHere, \(-1^{n+1}\arctan(n)\) is decreasing because \(\arctan(n)\) is a decreasing function. Also, the limit as \(n\) goes to infinity of \(\arctan(n)\) equals \(\pi/2\). So, the conditions are satisfied and thus the series is conditionally convergent.

Step 3: Test absolute convergence

To test if the series is absolutely convergent, we need to look at the series \(\sum |\arctan(n)|\). However, it's evident that this series actually diverges because of the Integral Test (the integral from \(1\) to \(\infty\) of \(|\arctan(x)|\) dx diverges). Therefore, the series doesn't converge absolutely.

Key Concepts

Alternating Series Test

When dealing with series, especially those with alternating signs, it's crucial to determine their convergence. The Alternating Series Test is an effective tool for analyzing such series. By definition, an alternating series takes the form \( \sum (-1)^n a_n \) where \( a_n \) is a sequence of positive terms.

To apply the test, check two conditions: first, the sequence \( a_n \) must be decreasing, and second, the limit of \( a_n \) as \( n \) approaches infinity must be zero. If both conditions are met, the series is said to converge. For the given series \( \sum_{n=1}^{\infty}(-1)^{n+1} \arctan \), due to the properties of the \(\arctan\) function, the conditions are satisfied, leading to the conclusion of conditional convergence.

It's important to note that while the Alternating Series Test can confirm convergence, it doesn't guarantee absolute convergence—another topic we'll dive into.

Absolute Convergence

Diving further into the realm of series convergence, absolute convergence refers to a stronger form of convergence. When we say a series \( \sum a_n \) is absolutely convergent, we imply that the series of absolute values \( \sum |a_n| \) also converges. This suggests that the series is less sensitive to the reordering of terms.

In the context of the given exercise, to examine absolute convergence, we consider the series \( \sum |(-1)^{n+1} \arctan(n)| \) which simplifies to \( \sum |\arctan(n)| \). However, as noted in the solution, this series does not converge since the integral of \( |\arctan(x)| \) over the interval from 1 to infinity is divergent. Consequently, the original alternating series does not meet the criteria for absolute convergence but is conditionally convergent.

Conditional Convergence

In contrast to absolute convergence, there is a phenomenon called conditional convergence. A series is conditionally convergent if it converges when its terms are taken with their original signs, but it does not converge absolutely. Essentially, the cancellation effect of the alternating signs plays a vital role in the convergence of the series.

In our exercise, the series \( \sum_{n=1}^{\infty}(-1)^{n+1} \arctan(n) \) is conditionally convergent because, while the alternating series test shows it converges, its corresponding series of absolute values diverges. It's a peculiar and interesting case where the alternating nature of the series contributes to its convergence, contrasting the behavior of its absolute counterpart.

Integral Test

The Integral Test serves as another pivotal tool in determining the convergence of series with non-negative terms. It connects the notion of an infinite series to that of an integral. If we're looking to apply this test, we can assess the convergence of \( \sum a_n \) by comparing it with the integral of \( f(x) \) from \( 1 \) to \( \infty \) where \( f(x) \) is a continuous, positive, and decreasing function that corresponds to the sequence \( a_n \) for all positive integers \( n \).

If the integral converges, so does the series; if the integral diverges, the series does too. In our exercise, to check for absolute convergence, we applied the Integral Test to the series \( \sum |\arctan(n)| \) and realized that the integral of \( |\arctan(x)| \) diverges. This led to the conclusion that the series does not converge absolutely, but as mentioned earlier, it does converge conditionally.