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. $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$$

Short answer

(a) Converges; (b) Converges; (c) Absolute convergence.

Step-by-step solution

Identify the Alternating Series

The given series \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}} \) is an alternating series. It can be written in the form \( \sum_{n=1}^{\infty} (-1)^{n+1} a_n \) where \( a_n = \frac{1}{n^2} \).

Check for Alternating Series Test

For an alternating series \( \sum (-1)^{n+1} a_n \) to converge, the sequence \( a_n \) must satisfy two conditions: it should be decreasing \( (a_n \geq a_{n+1}) \) and its limit as \( n \to \infty \) should be zero. The sequence \( a_n = \frac{1}{n^2} \) is decreasing since \( \frac{1}{n^2} > \frac{1}{(n+1)^2} \), and \( \lim_{n \to \infty} \frac{1}{n^2} = 0 \). Thus, the series converges.

Test Absolute Convergence

To check if the series \( \sum_{n=0}^{\infty}\left|a_{n}\right| \) (i.e., \( \sum_{n=1}^{\infty} \frac{1}{n^{2}} \)) converges, we recognize this as a p-series with \( p = 2 > 1 \). P-series \( \sum_{n=1}^{\infty} \frac{1}{n^{p}} \) converges if \( p > 1 \). Therefore, \( \sum_{n=1}^{\infty} \frac{1}{n^{2}} \) converges.

Determine Type of Convergence

Since the series \( \sum_{n=1}^{\infty} \frac{1}{n^{2}} \) converges and this is the absolute series of the alternating series, the original alternating series' convergence is absolute.

Key concepts

Alternating Series Test

An alternating series is a series whose terms alternate in sign. A classic example is of the form \[ \sum_{n=1}^{\infty} (-1)^{n+1} a_n \] where the sequence \( a_n \) consists of positive terms. This type of series can converge or diverge based on specific criteria.The Alternating Series Test helps determine the convergence of such series. This test requires:

• The sequence \( a_n \) should be decreasing, meaning \( a_n \geq a_{n+1} \).
• The limit of \( a_n \) as \( n \to \infty \) should be zero.

If both conditions are met, the alternating series converges. In our original exercise, we found that the sequence \( \frac{1}{n^2} \) satisfies these conditions, ensuring convergence.

P-Series

A p-series is a series of the form \[ \sum_{n=1}^{\infty} \frac{1}{n^p} \] where \( p \) is a real number. The convergence of a p-series is determined by the value of \( p \).A p-series converges if \( p > 1 \) and diverges for \( p \leq 1 \). The solution in our exercise highlights that \[ \sum_{n=1}^{\infty} \frac{1}{n^2} \] is a p-series with \( p = 2 \). Given that \( p > 1 \), this series converges. Understanding p-series is crucial when dealing with absolute convergence of alternating series.

Conditional Convergence

Conditional convergence occurs when a series \( \sum a_n \) converges, but its absolute series \( \sum |a_n| \) diverges. For a series to be considered conditionally convergent, the alternating series itself converges, but the series formed from taking the absolute values of its terms does not.In the exercise example, we're tasked to check if the alternating series' convergence is conditional or absolute. Since both the alternating series and its absolute version are found to converge, the series is not conditionally convergent. Instead, it has absolute convergence.

Absolute Convergence

A series \( \sum a_n \) is said to have absolute convergence when the series of the absolute values \( \sum |a_n| \) converges. This is a stronger form of convergence compared to conditional convergence.In our exercise, once the absolute series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) was checked and determined to converge (being a p-series with \( p = 2 \)), it confirmed the absolute convergence of the original alternating series.When a series is absolutely convergent, it implies that rearranging the terms of the series won't affect its sum, a property that only holds under absolute convergence. This characteristic sets it apart from conditionally convergent series, where reordering can change the sum.