Problem 1 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 1 through 9, determine whether the series converges or diverges. Explain your reasoning. $$ 1-2+3-\cdots+(-1)^{(n+1) n}+\cdots $$

Step-by-Step Solution

Answer

The given series is divergent.

Step 1: Identify the Series Type

Check if the given series is an alternating series. An alternating series alternates in sign from positive to negative or vice versa. Here, the nth term of the series is given by \((-1)^{(n+1)}n\). With \(n=1,2,3,4,\ldots\), the terms are 1, -2, 3, -4, alternating in sign, hence is an alternating series.

Step 2: Apply the Alternating Series Test

An alternating series converges if the limit of the absolute value of the terms is zero, and the magnitude of the terms are decreasing. In order to apply the alternating series test, divide it into two parts: the alternating part \((-1)^{(n+1)}\), and the term \(n\). We see that the sequence \(n\) increases as \(n\) gets larger, and therefore, the magnitude of the terms do not decrease as \(n\) increases. Thus, the alternating series test does not conclude anything about the convergence of the series.

Step 3: Divergence Test

Since alternating series test does not conclude anything, it's necessary to perform the divergence test. The divergence test states that the series \(\sum a_n\) diverges if the sequence \(a_n\) does not approach zero. Here, our sequence is \(n\), which obviously do not approach zero as \(n\) increases. Hence, by divergence test, the series diverges.

Key Concepts

Alternating Series Test

The alternating series test is a crucial tool in determining the convergence of series where the terms alternate in sign. An alternating series has the form \( (-1)^nf(n) \) or \( (-1)^{n+1}f(n) \), where \( f(n) \) is a sequence of positive terms. According to the alternating series test, such a series converges if two conditions are met:

• The absolute value of the terms \( f(n) \) must decrease monotonically, meaning each term is less than or equal to the previous term.
• The limit of the terms as \( n \) approaches infinity must be zero, formally written as \( \lim_{n \rightarrow \infty} f(n) = 0 \).

If these conditions aren't satisfied, the test is inconclusive; it doesn't necessarily mean the series diverges, but other methods must be used to determine its behavior.

Series Convergence or Divergence

Understanding whether a series converges or diverges is essential in analyzing mathematical sequences and functions. Convergence means that the series approaches a certain finite value as \( n \) tends to infinity, while divergence means the terms do not settle to a limit and may go to infinity or oscillate without reaching a particular value.

To ascertain convergence, a variety of tests can be applied, including the alternating series test, ratio test, root test, and comparison test, among others. Each test has its own set of conditions and is applicable to specific types of series. The challenge is choosing the appropriate test and correctly interpreting the results. Incorrect application may lead to wrong conclusions about a series' behavior.

Divergence Test

The divergence test, also called the nth-term test, offers a quick way to determine the divergence of a series. The divergence test states that if the limit of the terms \( a_n \) of a series \( \sum a_n \) does not equal zero, or the limit doesn't exist, then the series must diverge. In mathematical terms, \( \lim_{n \rightarrow \infty} a_n eq 0 \) implies that \( \sum a_n \) diverges.

It's important to note that this test can only be used to show divergence. If the limit of the terms is zero, the test is inconclusive, and the series may still diverge or converge. For conclusive proof of convergence, another test, like the alternating series test or ratio test, would then be required.