Question

Is the series convergent or divergent? If convergent, is it absolutely convergent? $$ \sum_{n=1}^{\infty} \sin \left(\frac{n \pi}{2}\right) $$

Short answer

The series is divergent.

Step-by-step solution

Analyze the Pattern of the Sine Function

The given series is \( \sum_{n=1}^{\infty} \sin \left(\frac{n \pi}{2}\right) \). First, calculate the first few terms of \( \sin \left(\frac{n \pi}{2}\right) \) to recognize a pattern. When \( n = 1 \), \( \sin \left(\frac{\pi}{2}\right) = 1 \). When \( n = 2 \), \( \sin \left(\pi\right) = 0 \). When \( n = 3 \), \( \sin \left(\frac{3\pi}{2}\right) = -1 \). When \( n = 4 \), \( \sin \left(2\pi\right) = 0 \). This pattern repeats: \( 1, 0, -1, 0 \).

Recognize the Sequence Pattern

The sequence \( \{1, 0, -1, 0, 1, 0, -1, 0, \ldots\} \) indicates that every four terms the sequence returns to its starting value. Thus, the series is alternating between \( 1 \) and \( -1 \) with zeros in between.

Analyze the Sum of the Series

Observe that the non-zero terms in the series are \( 1, -1, 1, -1, \ldots \). Except for the zero terms, the series effectively boils down to \( 1 - 1 + 1 - 1 + \ldots \). This sequence does not approach any single value, meaning the sum of such terms does not converge.

Conclusion on Convergence

Since the series neither approaches a specific value nor makes the partial sums stabilize at a certain number, it does not converge. Therefore, the given series is divergent. Absolute convergence is not applicable here as divergence is already confirmed.

Key concepts

Sine Function

Sine is a fundamental trigonometric function often encountered in mathematical series and equations. The sine function, denoted as \( \sin(x) \), varies in a predictable manner based on the angle \( x \). When dealing with series, observing the sine function can quickly reveal a repeating pattern. For example, the function \( \sin\left(\frac{n\pi}{2}\right) \) cycles through a sequence as \( n \) takes on natural number values.
In this particular series, when you calculate \( \sin\left(\frac{n\pi}{2}\right) \) for successive values of \( n \), a repeating pattern emerges: \( 1, 0, -1, 0 \). This cycle occurs every four terms. Such regularity helps in analyzing the convergence of the series. Recognizing the behavior of sine at specific intervals, like multiples of \( \frac{\pi}{2} \), is a critical skill when determining the characteristics of a series.

Alternating Series

An alternating series is a series in which the terms alternate in sign. This means the sequence includes positive and negative terms. For the series \( \sum_{n=1}^{\infty} \sin\left(\frac{n\pi}{2}\right) \), we see a pattern of alternating terms: \( 1, 0, -1, 0, 1, 0, -1, 0, \ldots \). Though the sequence also contains zeroes, the non-zero elements switch between positive and negative.
This alternating nature affects the analysis of convergence. In general, an alternating series converges if the absolute value of the terms decreases monotonically to zero. However, in this series, the recurring numerical pattern affects convergence analysis, as further explained in the next section.

Divergence in Series

Divergence in series refers to a series whose partial sums do not tend to a single finite limit. It essentially means the series does not "settle down" to a particular value. In our series with sine function: \( \sum_{n=1}^{\infty} \sin\left(\frac{n\pi}{2}\right) \), the non-zero terms form a repeating sequence: \( 1, -1, 1, -1, \ldots \).
The issue is that while zeros separate the alternating ones and negative ones, the sum of non-zero terms never stabilizes. The partial sums oscillate between values such as \( 1, 0, 1, 0, \ldots \). Each addition cancels out the previous, preventing the series from converging. If a series does not converge, it diverges, implying here that it cannot sum to a finite number and, therefore, is not absolutely convergent. Understanding divergence is key in recognizing when a mathematical series doesn't settle into a predictable pattern.