Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n-1}{n} $$

Step-by-Step Solution

Answer

The series is divergent.

Step 1: Identify the Series Type

The given series is \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n-1}{n} \). This is an alternating series because of the factor \((-1)^{n+1}\).

Step 2: Check Alternating Series Test Conditions

For the alternating series \( \sum (-1)^{n+1} b_n \), it will converge if \( b_n \to 0 \) as \( n \to \infty \) and \( b_{n+1} \leq b_n \) for all \( n \). Here, \( b_n = \frac{n-1}{n} \).

Step 3: Check Convergence of \( b_n \)

As \( n \to \infty \), \( b_n = \frac{n-1}{n} \to 1 \). Since \( b_n \) does not approach 0, the alternating series test cannot be used to show convergence.

Step 4: Absolute Convergence Test

A series \( \sum a_n \) is absolutely convergent if \( \sum |a_n| \) converges. Check if \( \sum_{n=1}^{\infty} \left| (-1)^{n+1} \frac{n-1}{n} \right| = \sum_{n=1}^{\infty} \frac{n-1}{n} \) converges.

Step 5: Determine Divergence of \( \sum \frac{n-1}{n} \)

The series \( \sum \frac{n-1}{n} \) simplifies to \( \sum \left(1 - \frac{1}{n}\right) \). This diverges, as it is a Harmonic series minus a divergent constant series.

Step 6: Conclusion on Convergence

Since the series fails the alternating series test and is not absolutely convergent (as the absolute value series diverges), it is divergent.

Key Concepts

Series Convergence

Understanding series convergence is crucial for analyzing whether a series sums up to a finite value as its terms extend indefinitely. A series is said to converge when its sequence of partial sums approaches a certain limit. Convergence can be tricky, as some series might seem to approach a sum but never really settle to a single value.
When evaluating convergence, look at the general behavior of the terms:

• If the terms do not tend to zero, the series diverges.
• If the terms do tend to zero but at a slow rate, further tests may be needed.

Determining convergence helps mathematicians classify series into different types, including absolute and conditional convergence.

Absolute Convergence

A series is absolutely convergent if the series formed by taking the absolute values of its terms also converges. To check for absolute convergence, consider \[ \sum |a_n| \].
This allows for a sharper lens on the series, assessing its behavior independently of sign changes. Absolute convergence implies convergence of the original series, but not vice versa.

• If \( \sum |a_n| \) converges, then \( \sum a_n \) converges absolutely.
• Absolute convergence is a strong form of convergence that guarantees regular convergence.

It's a key concept because once a series is found to be absolutely convergent, it can be rearranged or manipulated freely without affecting its convergence.

Harmonic Series

The harmonic series is a well-known series in mathematics defined as \[ \sum_{n=1}^{\infty} \frac{1}{n} \].
Despite the terms decreasing to zero, the harmonic series diverges. This means that its partial sums do not approach a finite limit.
It serves as a classic example of a divergent series, showing that even series that 'seem' to lessen their sums over time may still not converge.

• This series diverges because the sum grows without bound.
• It's often compared to other series to test for divergence.

Understanding the harmonic series helps in identifying behavior in other complex series by pointing out that decreasing terms alone aren't enough for convergence.

Alternating Series Test

The alternating series test is helpful for determining the convergence of series that alternate in sign. An alternating series takes the form \[ \sum_{n=1}^{\infty} (-1)^n b_n \] or \[ \sum_{n=1}^{\infty} (-1)^{n+1} b_n \].
For this test to confirm convergence, two conditions must be met:

• The terms \( b_n \) must tend to zero as \( n \to \infty \).
• The sequence \( b_n \) must be decreasing, i.e., \( b_{n+1} \leq b_n \) for all \( n \).

If both conditions are satisfied, the series converges. However, failing either condition means the test cannot confirm convergence. Understanding and applying this test is crucial for evaluating the convergence of alternating series, as it provides clear criteria under which convergence can be guaranteed.