Problem 17 - Essential Calculus

Question

Explain why the Integral Test does not apply to the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n} $$

Step-by-Step Solution

Answer

The reason the Integral Test cannot be applied to the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\) is because the function \(f(n) = \frac{(-1)^{n}}{n}\) is not positive on the interval \([1, \infty)\). The Integral Test requires that the function \(f(n)\) be positive, continuous, and decreasing on this interval. In this case, the function \(f(n)\) fails the positivity condition, because it alternates between positive and negative values for different \(n\).

Step 1: Identify the sequence term function

Looking at the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\), it can be determined that the sequence term function \(f(n)\) is \(\frac{(-1)^{n}}{n}\).

Step 2: Test for Continuity

The function \(f(n)\) is continuous for all \(n > 0\), except for \(n = 0\), since the function is not defined at \(n = 0\). But since our region of interest is \(n \geq 1\), we can ignore it. Thus, the function \(f(n)\) is continuous on \([1, \infty)\).

Step 3: Test for positivity

The function \(f(n)\) is not positive for all \(n \geq 1\). Instead, it alternates between positive and negative values, because of the presence of \((-1)^n\) term. Thus, the function \(f(n)\) is not positive on the interval \([1, \infty)\).

Step 4: Test for monotonic decreasing

Although the magnitude of terms in the sequence |\(f(n)\)| = \(\frac{1}{n}\) is decreasing, \(f(n)\) itself does not consistently decrease because it alternates in sign.

Key Concepts

Alternating Series

An alternating series is a series where the signs of its terms alternate between positive and negative. This is exactly what we observe in the series given:

• The term \((-1)^{n}\) alternates between -1 and 1.
• As a result, each term switches its sign.

For example, the series \[\sum_{n=1}^{fty} \frac{(-1)^{n}}{n}\]generates terms like \(-1, \frac{1}{2}, -\frac{1}{3}, \frac{1}{4}, \ldots\)
Alternating series are important in calculus because they offer unique patterns of convergence. To determine if an alternating series converges, we use the Alternating Series Test, which is separate from the Integral Test mentioned.
The Alternating Series Test states that a series \(\sum \frac{(-1)^{n}a_{n}}{n}\) converges if:

• The absolute value of the terms \(|a_n|\) is decreasing.
• The limit as \(n\) approaches infinity of \(a_n\) is zero.

If these conditions are met, the alternating series converges.

Sequence

A sequence is a function \(f(n)\) on a set of natural numbers that assigns values in a specific order. In the context of series, a sequence refers to the individual terms that are summed.
For our series\[\sum_{n=1}^{fty} \frac{(-1)^{n}}{n}\]
The sequence is \(\left\{\frac{(-1)^{n}}{n}\right\}\) leading to sums of numbers alternating between positive and negative.

• The sequence starts at \(-1\), then \(\frac{1}{2}\), then \(-\frac{1}{3}\).
• It continues this pattern indefinitely.

Sequences are the building blocks of series. Each individual term's behavior plays a vital role in determining whether the whole series converges or diverges.

Convergence

Convergence of a series refers to the condition where the sum of its infinite terms approaches a fixed value as more terms are added.
In simpler words, if we have a series, and as we keep adding more terms, the total sum does not keep increasing or decreasing indefinitely but settles towards a specific number, this series is convergent.
With the series \[\sum_{n=1}^{fty} \frac{(-1)^{n}}{n}\]
We explore various tests to check for convergence. However, shown in the solution, the Integral Test does not apply since:

• The series' terms are not all positive, violating an integral test condition.

Yet, the Alternating Series Test confirms its convergence.
Understanding convergence helps in grasping how infinite series can represent finite sums, a fundamental concept in calculus.

Mathematical Function

In calculus, a function is a relation that assigns exactly one output for each input. The term "mathematical function" can be applied to sequences or series to define the rule of generating terms.
Considering the sequence term function \(f(n) = \frac{(-1)^{n}}{n}\), derived from the series\[\sum_{n=1}^{fty} \frac{(-1)^{n}}{n}\]It highlights how the sign and values change at each integer \(n\):

• This function is defined on \(\mathbb{N}\) except \(n=0\) and hence considered continuous for \(n \geq 1\).
• It exhibits alternating behavior due to \((-1)^{n}\).

Mathematical functions play a pivotal role in analyzing the behavior of sequences and series in terms of continuity, positivity, and monotonicity. These properties help apply suitable tests for convergence.