Convergence of Alternating Series Sum 1nsqrtn

Given a series $\sum a_n$, determine whether it converges absolutely or conditionally. Specifically, consider $\sum (-1)^{n-1}/\sqrt{n}$ and evaluate its convergence using the alternating series test and discuss its absolute convergence.

In this problem, we explore the convergence properties of an alternating series, particularly focusing on whether the series converges absolutely or conditionally. The series in question involves terms of the form $(-1)^{n-1}/\sqrt{n}$, which suggests the application of the alternating series test—a tool useful for evaluating the convergence of series whose terms alternate in sign.

The test provides conditions under which an alternating series is convergent. Specifically, if the absolute value of the terms decreases monotonically to zero, the series converges. Applying this to our series shows that it does indeed converge as the terms decrease and approach zero over time; however, there is more to consider. This leads us to the concept of absolute convergence. A series is absolutely convergent if the series formed by taking the absolute values of its terms also converges. If a series is absolutely convergent, it is convergent in the regular sense as well.

For our series, taking absolute values gives us $1/\sqrt{n}$, which is a p-series with p = 1/2. Since p is less than 1, this series is divergent. Therefore, although the original series is conditionally convergent by the alternating series test, it does not converge absolutely. Analyzing series in this manner speaks to broader concepts used in real analysis, such as comparing rates of divergence and the behavior of positive term series in contrast to alternating ones.