Question

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

Short answer

The series does not converge.

Step-by-step solution

Analyze the general term

The general term of the series is given by \(a_n = \sin \left( \frac{n \pi}{2} \right) \sin \left( \frac{1}{n} \right)\). We need to understand the behavior of this term as \(n\) changes.

Simplify the problem using trigonometric values

Since \(\sin \left(\frac{n\pi}{2}\right)\) takes on the values 0, 1, 0, and -1 periodically depending on whether \(n \equiv 0, 1, 2, 3 \pmod{4}\), we simplify the series into subseries based on these periodic terms.

Consider terms induced by periodic oscillations

The series becomes non-zero only when \(n \equiv 1, 3 \pmod{4}\). For these values, \(\sin(\frac{n\pi}{2})\) is \(1\) or \(-1\), which simplifies the terms to \(\sin \left( \frac{1}{n} \right)\) or \(-\sin \left( \frac{1}{n} \right)\).

Analyze convergence of simplified terms

The value of \(\sin \left( \frac{1}{n} \right)\) approximates \(\frac{1}{n}\) for large \(n\). Hence, the problem reduces to checking the convergence of \(\sum_{n=1, n \equiv 1, 3 \pmod{4}}^{\infty} \frac{1}{n}\).

Apply the convergence test

The series \(\sum \frac{1}{n}\), known as the harmonic series, is divergent. Because the terms \(\frac{1}{n}\) appear periodically and are equivalent to the harmonic series for the specific subsequences, the subseries is also divergent.

Key concepts

Conditional Convergence

Conditional convergence occurs when a series converges, but the series of absolute values does not. Consider a series \(\sum a_n\\). A series is conditionally convergent if \(\sum a_n\\) converges but \(\sum |a_n|\\) does not. This concept is essential because it shows how a series can converge even if the absolute values of its terms do not. Conditional convergence delicately balances terms both below and above zero, allowing partial sums to gradually settle towards a limit.

• Consider the alternating harmonic series \(\sum (-1)^{n+1}/n\\): it converges conditionally.
• The series \(\sum 1/n\\) diverges, but the alternating versions do not.

Recognizing whether a series is conditionally convergent can be challenging, often requiring careful rearrangement or known convergence tests like the Alternating Series Test. Understanding this concept lays groundwork for absolute convergence.

Absolute Convergence

A series \(\sum a_n\\) converges absolutely if the series of absolute values \(\sum |a_n|\\) also converges. Absolute convergence implies convergence unconditionally. This means that even upon any rearrangement of terms, the series will still converge, unlike conditionally convergent series.

• An absolutely convergent series showcases stable convergence outcomes.
• If \(\sum |a_n|\\) converges, then \(\sum a_n\\) definitely converges as well; however, the reverse isn't always true.

Understanding absolute convergence ensures stability in computation and proves particularly useful when applying the Limit Comparison Test or the Ratio Test. Absolute convergence helps diagnose how individual terms influence the series' overall behavior.

Harmonic Series

The harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\\) is one of the most famous examples in mathematical series studies. It diverges, meaning it grows without bound as more terms are added. Despite each term shrinking, the total never settles at a limit.

• The divergence is a critical concept demonstrating how slowly decreasing sequences can still sum to infinity.
• The harmonic series is pivotal when comparing with other series using the Comparison Test.

Despite its divergence, variations such as the alternating harmonic series exhibit unique properties in convergence. Recognizing the harmonic series' behavior aids in understanding more complex series types and grasping critical divergence characteristics.

Trigonometric Series

Trigonometric series generalize the concept of series by incorporating sine and cosine terms. These series can represent periodic functions using plain waves:

• They are particularly important in Fourier Analysis for expressing functions as sums of sine and cosine.
• In the given original exercise, periodic properties of \(\sin(n\pi/2)\\) simplify the series.

Analyzing a trigonometric series involves understanding both periodicity and amplitude changes over time, which are crucial when determining convergence. The insight gained from trigonometric series helps solve problems associated with waves, vibrations, and signals. Leveraging symmetries and repeating patterns found in trigonometric series allows for series simplification, as seen when decomposing and analyzing the given series.