Question 8.5.3: Testing for Absolute Convergence Determine whether ∑^∞k=1 si...
Testing for Absolute Convergence
Determine whether \sum_{k=1}^{\infty} \frac{\sin k}{k^3} is convergent or divergent.
Notice that while this is not a positive-term series, neither is it an
alternating series. Because of this, our only choice is to test the series for absolute convergence. From the graph of the first 20 partial sums seen in Figure 8.35, it appears that the series is converging to some value around 0.94. To test for absolute convergence, we consider the series of absolute values, \sum_{k=1}^{\infty}\left|\frac{\sin k}{k^3}\right|. Notice that
\left|\frac{\sin k}{k^3}\right|=\frac{|\sin k|}{k^3} \leq \frac{1}{k^3}, (5.2)
since |sin k| ≤ 1, for all k. Of course, \sum_{k=1}^{\infty} \frac{1}{k^3} is a convergent p-series (p = 3 > 1). By the Comparison Test and (5.2), \sum_{k=1}^{\infty}\left|\frac{\sin k}{k^3}\right|. converges, too. Consequently, the original series \sum_{k=1}^{\infty} \frac{\sin k}{k^3} converges absolutely and hence, converges.
