Question

State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty} \frac{1}{n} \sin \left(\frac{n \pi}{2}\right)\)

Short answer

The series converges conditionally.

Step-by-step solution

Recognize the series' general term

The general term of the series given is \(a_n = \frac{1}{n} \sin\left(\frac{n \pi}{2}\right)\). To analyze the convergence, we first need to understand the behavior of \(\sin\left(\frac{n \pi}{2}\right)\) for different integer values of \(n\).

Evaluate the sine term for integers

The sine function \(\sin\left(\frac{n \pi}{2}\right)\) has specific values based on the modulo of \(n\) with 4: it equals 0 when \(n\%4 = 0\) or \(n\%4 = 2\); -1 when \(n\%4 = 3\); and 1 when \(n\%4 = 1\). Thus, the series effectively has non-zero terms only when \(n\%4 = 1\) or \(n\%4 = 3\).

Simplify the series

Given the pattern identified, the series can be rewritten as: \(\sum_{n=0}^{\infty} \left(\frac{1}{4n+1} - \frac{1}{4n+3}\right)\). Here, we grouped the positive and negative terms separately.

Consider the Absolute Convergence Test

Since the terms alternate, consider the absolute values \( \sum_{n=1}^{\infty} \left|a_n\right| = \sum_{n=0}^{\infty} \left(\frac{1}{4n+1} + \frac{1}{4n+3}\right)\). This resembles the harmonic series, which diverges. Hence, the series does not converge absolutely.

Apply the Alternating Series Test for Conditional Convergence

The alternating series \(\sum_{n=0}^{\infty} \left(\frac{1}{4n+1} - \frac{1}{4n+3}\right)\) satisfies the conditions for the alternating series test: the terms \(\frac{1}{4n+1}\) and \(\frac{1}{4n+3}\) decrease in magnitude and approach zero. Thus, it converges conditionally.

Key concepts

alternating series test

The Alternating Series Test is a handy tool when analyzing series that exhibit alternating positive and negative terms. For a series \(\sum_{n=1}^{\infty} (-1)^{n} b_n\), it can be used to determine whether these series converge. There are two main criteria:

• The absolute values of the terms, \( b_n \), must decrease monotonically, meaning that each term is smaller than the previous one.
• The limit of \( b_n \) as \( n \) approaches infinity must be zero: \( \lim_{{n \to \infty}} b_n = 0 \).

If both conditions are met, the series converges. This tool is particularly effective for the alternating series we often encounter in calculus exercises. For example, the series given in the exercise, \(\sum_{n=0}^{\infty} \left(\frac{1}{4n+1} - \frac{1}{4n+3}\right)\), was determined using this test to confirm conditional convergence.

harmonic series

The Harmonic Series is one of the most well-known divergent series in mathematics. It can be expressed as:
\[\sum_{n=1}^{\infty} \frac{1}{n}\]
This series is famous for its divergence, meaning that it increases without bounds as more terms are added. Even though the terms become smaller and smaller, they do not decrease quickly enough for the series to converge.

In the context of the exercise, the series resembles the harmonic series when considering absolute values. Understanding that the harmonic series diverges was key in knowing that the transformed absolute series also diverges, ruling out absolute convergence.

absolute convergence

Absolute Convergence occurs when the series formed by taking the absolute value of its terms converges. For example, given a series \( \sum_{n=1}^{\infty} a_n \), if \( \sum_{n=1}^{\infty} |a_n| \) converges, then the original series also converges absolutely.

This means that convergence is unaffected by the signs of the terms. If a series converges absolutely, it converges under all circumstances, making it a stronger form of convergence than conditional convergence.

In the provided exercise, it was determined that when considering the absolute values of the series terms, it behaves like a harmonic series, which diverges. Thus, this series does not meet the criteria for absolute convergence.

conditional convergence

Conditional Convergence is a unique type of convergence where a series converges only because of the cancellation effect produced by alternating signs. The series converges overall, but the series of absolute values diverges.

This is different from absolute convergence, which requires the series of absolute values to converge as well. The Alternating Series Test is often used to determine conditional convergence.

The example from the exercise is a perfect illustration of conditional convergence. The series \( \sum_{n=0}^{\infty} \left(\frac{1}{4n+1} - \frac{1}{4n+3}\right) \) fulfills the alternating series test's criteria, so it converges, but only in a conditional manner. This means that it's the alternating nature that allows it to converge, despite its absolute counterpart diverging.