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. $$\sum_{n=1}^{\infty} \cos (\pi n)$$

Short answer

The series diverges, both originally and absolutely, thus there is no conditional or absolute convergence.

Step-by-step solution

Identify the Terms of the Series

For the series \(\sum_{n=1}^{\infty} \cos(\pi n)\), we first examine \(\cos(\pi n)\). When \(n\) is an integer, \(\cos(\pi n)\) alternates between \(-1\) and \(1\). Specifically, \(\cos(\pi) = -1\), \(\cos(2\pi) = 1\), etc. Hence, each term \(a_n = \cos(\pi n)\) of the series is an oscillating sequence of \(-1, 1, -1, 1, \ldots\). This irregular behavior suggests divergence.

Test for Convergence or Divergence of the Original Series

For the series \(\sum_{n=1}^{\infty} \cos(\pi n)\), observe that the terms do not approach zero as \(n\) goes to infinity. The terms alternate between \(-1\) and \(1\), neither approaching zero. According to the Divergence Test (or the \(\lim_{n \to \infty} a_n eq 0\) test), if terms do not approach zero, the series diverges. Therefore, this series diverges.

Determine Convergence of the Absolute Series

Now consider the series \(\sum_{n=1}^{\infty} |\cos(\pi n)|\). Since \(|\cos(\pi n)| = 1\) for all integer \(n\), the series becomes \(\sum_{n=1}^{\infty} 1\). This is a series where every term is \(1\), so the partial sums grow without bound, indicating that this series diverges.

Assess Conditional vs Absolute Convergence

Since \(\sum_{n=1}^{\infty} a_n\) (the original series) diverges, we don't need to assess conditional vs absolute convergence according to the given problem's typical order of operations. Normally, a series needs to converge for this assessment. Here, both the original and absolute series diverge. Therefore, there's no convergence to classify as conditional or absolute.

Key concepts

Convergence and Divergence

When examining a series, one of the core tasks is to determine whether it converges or diverges. This can tell us a lot about the behavior and properties of the series. For convergence, intuitively, it's when the sum of the terms doesn't keep increasing indefinitely. Instead, it approaches a specific limit as more terms are added.
In contrast, divergence means the sum keeps getting larger (or smaller), and there's no fixed number that it settles down to. For alternating series, which have terms that switch in sign, convergence isn't always straightforward. If the terms do not get closer and closer to zero, as is the case in our exercise with "cos(πn)" (constantly oscillating between -1 and 1), the series diverges.

• Key indicator of divergence: terms not approaching zero.
• Convergence means the sum stabilizes to a particular value.
• For alternating series, special checks are needed for convergence.

Absolute Convergence

Absolute convergence deals with the idea that if you take the absolute value of each term in the series, does the resulting series converge? This provides a stronger condition than regular convergence, and if a series is absolutely convergent, it is also simply convergent. In the exercise, the series formed by the absolute values of the terms becomes \[\sum_{n=1}^{\infty} |\cos(\pi n)| = \sum_{n=1}^{\infty} 1,\]which is basically an infinite sum of "1's." Naturally, this series diverges because the sum keeps increasing without bound, showing **no absolute convergence** here.

• Absolute convergence is stricter than normal convergence.
• \(\sum_{n=1}^{\infty} |a_n|\) must converge for absolute convergence.
• If absolutely convergent, then the series is also (simply) convergent.

Conditional Convergence

Conditional convergence applies to series that converge only when considering their original form but not when using the absolute values of their terms. It is a finer detail of convergence that many students find tricky but essential in deeper analysis of series. For example, the alternating harmonic series \(\sum_{n=1}^{\infty} \frac{(-1)^n}{n}\)is conditionally convergent because it converges normally but diverges when you take the absolute value of its terms.
In our exercise, though, both the original and absolute series diverge. Thus, there is no convergence to explore under this category. When handling real-world problems or complicated calculations, identifying such detailed behaviors in a series can be crucial in understanding their long-term behavior.

• Applicable after confirming normal convergence but divergence in absolute terms.
• Conditional convergence is a nuanced concept in detailed analyses.
• No need to determine here as the series does not converge ordinarily.