Convergence of Alternating Series

Take the series $(-1)^{n-1} B_n$, where $B_n$ is positive. Determine if the series is convergent or divergent based on if $B_{n+1} \leq B_n$ and $\lim_{{n \to \infty}} B_n = 0$.

In this problem, you are dealing with an alternating series of the form $(-1)^{n-1} B_n$, where $B_n$ is a sequence of positive terms. This specific type of series is known for its alternating signs, and understanding its convergence hinges on particular criteria known as the Alternating Series Test. This test is a valuable tool in your mathematical toolkit as it simplifies the analysis of such series by focusing primarily on the behavior of the sequence $B_n$.

The Alternating Series Test states that if the sequence of positive terms, $B_n$, is decreasing, and as $n$ approaches infinity, the terms $B_n$ tend to zero, then the alternating series converges. This means you need to verify two critical conditions: the monotonic decrease in the sequence $B_n$ and its limit towards zero. These conditions help us decide without having to find the sum directly, which can often be cumbersome.

Understanding this test not only aids in determining the convergence or divergence of alternating series but also provides insight into the nature of infinite series in general. It's a stepping stone to deeper concepts, such as absolute and conditional convergence, which further explore the series' properties when stripped of its alternating signs. By mastering these concepts, you'll enhance your ability to tackle a wide array of problems involving series in calculus and beyond.