Question

An alternating series \(\sum_{n=i}^{\infty} a_{n}\) is given. (a) Determine if the series converges or diverges. (b) Determine if \(\sum_{n=0}^{\infty}\left|a_{n}\right|\) converges or diverges. (c) If \(\sum_{n=0}^{\infty} a_{n}\) converges, determine if the convergence is conditional or absolute. $$\sum_{n=2}^{\infty} \frac{\sin ((n+1 / 2) \pi)}{n \ln n}$$

Short answer

(a) The series converges. (b) The absolute series diverges. (c) Convergence is conditional.

Step-by-step solution

Identify the sequence

The given series is \( \sum_{n=2}^{\infty} \frac{\sin((n+1/2)\pi)}{n \ln n} \). We know that \( \sin((n+1/2)\pi) = (-1)^{n} \). Therefore, the series becomes \( \sum_{n=2}^{\infty} \frac{(-1)^{n}}{n \ln n} \).

Apply Alternating Series Test

To determine convergence, check if the Alternating Series Test applies. For the series \( \sum (-1)^{n} a_n \), it converges if the absolute value of the terms \( a_n = \frac{1}{n \ln n} \) is decreasing and has a limit of zero. - \( \frac{1}{n \ln n} \) is decreasing for \( n \geq 2 \) because for \( n \ln n \), it gets larger as \( n \) increases.- \( \lim_{n \to \infty} \frac{1}{n \ln n} = 0 \) as the denominator grows much faster than the numerator.

Consider Convergence of Absolute Series

Check if \( \sum_{n=2}^{\infty} \left|\frac{(-1)^{n}}{n \ln n}\right| = \sum_{n=2}^{\infty} \frac{1}{n \ln n} \) converges.- This is similar to the \( p\)-series \( \sum_{n=2}^{\infty} \frac{1}{n^{1+\epsilon}} \), but here the factor \( \ln n \) makes it less than \( p = 1 \).- We use the integral test on \( \int_{2}^{\infty} \frac{1}{x \ln x} \, dx \) which diverges. Thus, \( \sum_{n=2}^{\infty} \frac{1}{n \ln n} \) diverges.

Determine Conditional or Absolute Convergence

Since the original series converges by the Alternating Series Test but the absolute series diverges, the convergence is conditional.

Key concepts

Convergence

Convergence in the context of series, particularly alternating ones, refers to the behavior of the series as it progresses to an infinite number of terms. For an alternating series like \( \sum_{n=2}^{\infty} \frac{(-1)^{n}}{n \ln n} \), convergence means that the sum of all terms approaches a finite number as more terms are added. The Alternating Series Test tells us that this series converges if two conditions are met: the sequence \( a_n = \frac{1}{n \ln n} \) must be decreasing, and the limit as \( n \) approaches infinity must equal zero. For this specific series, both conditions are satisfied, confirming convergence.

Divergence

In contrast, divergence means that as more terms are added to a series, the sum does not stabilize but instead grows indefinitely or fluctuates without settling to a single value. For the series \( \sum_{n=2}^{\infty} \frac{1}{n \ln n} \), derived from taking the absolute value of the original series terms, the series diverges. This conclusion was reached using the integral test, which showed that the integral \( \int_{2}^{\infty} \frac{1}{x \ln x} \, dx \) diverges. This outcome indicates that the series does not sum up to a finite number as \( n \) tends to infinity.

Absolute Convergence

Absolute convergence refers to the situation where a series and its associated series of absolute values both converge. This is a stronger form of convergence because it indicates stability in the sum regardless of sign changes in the series terms. However, for the series \( \sum_{n=2}^{\infty} \frac{(-1)^{n}}{n \ln n} \), absolute convergence is not present. This is because its corresponding absolute series \( \sum_{n=2}^{\infty} \frac{1}{n \ln n} \) diverges, showing that taking the absolute value alters the series just enough to lose its finite-sum property.

Conditional Convergence

Conditional convergence occurs when a series converges but would not if all its terms were made positive. This happens in our alternating series \( \sum_{n=2}^{\infty} \frac{(-1)^{n}}{n \ln n} \) which converges by the Alternating Series Test. Meanwhile, its absolute version \( \sum_{n=2}^{\infty} \frac{1}{n \ln n} \) diverges. Therefore, the convergence of the alternating series is conditional. This type of convergence highlights the delicate balance that alternating terms provide, which, when removed by absolute values, results in divergence.