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An absolutely convergent series is one where the series of the absolute values of the terms converges. This means if we take all the negative signs away, and the series still converges, then it's absolutely convergent. This is a strong form of convergence. To determine if a series is absolutely convergent, we examine the absolute sum \[ \sum_{k=1}^{\infty} \left| a_k \right| \].

In our original series \( \sum_{k=1}^{\infty} \frac{\cos k}{k^{3}} \), we look at \( \sum_{k=1}^{\infty} \left|\frac{\cos k}{k^{3}}\right| \). Since \(|\cos k| \leq 1\), we can compare this to the series \( \sum_{k=1}^{\infty} \frac{1}{k^{3}} \). This series \( \frac{1}{k^{3}} \) is a p-series with \( p = 3 \), and since \( p > 1 \), the series converges. Hence, the original series is absolutely convergent.

While the given problem determines that the series is absolutely convergent, understanding what conditional convergence is can be helpful in many other contexts. A series is conditionally convergent if it converges only when you consider the original terms, not their absolute values. In simpler words, remove the negative signs, and it becomes divergent.

Mathematically, a series \( \sum_{k=1}^{\infty} a_k \) is conditionally convergent if the series itself converges, but the series of absolute values \( \sum_{k=1}^{\infty} \left| a_k \right| \) diverges.

For the series \( \sum_{k=1}^{\infty} \frac{\cos k}{k^{3}} \), the determination of its absolute convergence meant skipping this check, because absolutely convergent series are naturally convergent as well. This is something you always need to remember: absolute convergence implies regular convergence, but not vice versa.

The comparison test is a handy tool for determining the convergence of a series. It involves comparing the series in question to another series that we already know converges or diverges.

To use this test, we find a series \( \sum_{k=1}^{\infty} b_k \) such that each term \( \left| a_k \right| \leq b_k \). If this comparison series converges and the terms are all non-negative, then the original series also converges. Similarly, if \( b_k \) diverges and \( a_k \geq b_k \) for large \( k \), our original series diverges.

In the problem, to evaluate absolute convergence, the absolute series \( \sum_{k=1}^{\infty} \left|\frac{\cos k}{k^{3}}\right| \) was compared to \( \sum_{k=1}^{\infty} \frac{1}{k^{3}} \), a known convergent p-series. Because \(|\cos k| \leq 1\), we know \( \left|\frac{\cos k}{k^{3}}\right| \leq \frac{1}{k^{3}} \), confirming the convergence of the original series.