Problem 25 - Thomas Calculus

Question

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1+n}{n^{2}} $$

Step-by-Step Solution

Answer

The series converges conditionally by alternating series test but not absolutely.

Step 1: Understanding Absolute Convergence

To determine absolute convergence, we consider the absolute value of the terms of the series: \[ \sum_{n=1}^{\infty} \left| (-1)^{n+1} \frac{1+n}{n^{2}} \right| = \sum_{n=1}^{\infty} \frac{1+n}{n^{2}}\] This simplifies to considering the series: \[ \sum_{n=1}^{\infty} \frac{1+n}{n^{2}} = \sum_{n=1}^{\infty} \left( \frac{1}{n^2} + \frac{1}{n} \right) \] This series is the sum of two series.

Step 2: Testing Absolute Convergence with Comparison

Break down the absolute series:\[ \sum_{n=1}^{\infty} \frac{1}{n^2} + \sum_{n=1}^{\infty} \frac{1}{n} \] The series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) is a convergent p-series (p = 2 > 1).However, the series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is a harmonic series and it diverges. Since part of the absolute series diverges, the entire series \( \sum_{n=1}^{\infty} \frac{1+n}{n^{2}} \) diverges.

Step 3: Testing for Conditional Convergence

Since the series does not converge absolutely, check for conditional convergence using the alternating series test:\[ a_n = \frac{1+n}{n^2} \]1. \( a_n \) is positive and decreasing: As \( n \to \infty \), \( \frac{1+n}{n^2} \to 0 \). Hence, the terms decrease to 0.2. Alternating nature: The term \((-1)^{n+1}\) alternates the series.Thus, the series converges by the alternating series test.

Key Concepts

Absolute Convergence

Absolute convergence in a series occurs when the series still converges even after taking the absolute value of all its terms. To check for absolute convergence, consider the modified series where each term's sign is ignored.
In our given example, the series is:

• \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1+n}{n^{2}} \)
• To check absolute convergence, we convert it to \( \sum_{n=1}^{\infty} \frac{1+n}{n^{2}} \)

Breaking this into two parts: \( \sum_{n=1}^{\infty} \frac{1}{n^{2}} \) and \( \sum_{n=1}^{\infty} \frac{1}{n} \), we can analyze each separately.
We find that the series \( \sum_{n=1}^\infty \frac{1}{n^2} \) converges because it is a p-series with \( p = 2 > 1 \). However, \( \sum_{n=1}^\infty \frac{1}{n} \) diverges as it is the harmonic series. Since only part of the series converges, the absolute series as a whole does not converge.
If any component of the series diverges, absolute convergence is not possible, leading to the conclusion that the original series does not converge absolutely.

P-Series

p-series is a type of mathematical series written in the form: \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). Whether a p-series converges depends on the value of \( p \):

• If \( p > 1 \), the series converges.
• If \( p \leq 1 \), the series diverges.

In our analysis of the series within the exercise, we identify \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) as a p-series. Here, \( p=2 \), which is greater than 1, ensuring convergence.
On the other hand, the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) can be seen as a p-series with \( p=1 \), and as our guidelines suggest, this series diverges.
Understanding these properties of p-series helps evaluate the convergence characteristics of more complex series through component analysis.

Alternating Series Test

The Alternating Series Test (sometimes called the Leibniz test) is a tool used to determine if an infinite alternating series converges. An alternating series takes the form: \( \sum_{n=1}^{\infty} (-1)^n a_n \). Two main conditions must be fulfilled:

• The sequence \( a_n \) must be positive, decreasing, and ultimately approach zero as \( n \) becomes very large.
• The series must alternate between positive and negative terms.

For our exercise, the alternating component is \((-1)^{n+1}\), and \( a_n = \frac{1+n}{n^{2}} \).
Firstly, we check that the terms \( a_n \) are positive and decreasing. As \( n \) increases, \( \frac{1+n}{n^2} \) indeed approaches zero, affirming the series progresses in a diminishing fashion. Secondly, the term structure confirms the alternating nature, fulfilling the test's requirements.
Hence, even though the series does not converge absolutely, it converges conditionally based on the alternating series test.