Exploring the Alternating Series Test: Convergence and Divergence

The Alternating Series Test is a method used to determine the convergence of a series with alternating positive and negative terms. The general form of an alternating series is: \(\sum_{n=1}^{\infty} (-1)^n a_n\)

where \(a_n\) is a sequence of positive terms.

The test has two key conditions for convergence:

1. Decreasing terms: The sequence \(a_n\) must be decreasing, meaning each term is smaller than the previous one for all sufficiently large \(n\). In other words, \(\leq a_n\).
2. Limit approaching zero: The limit of \(a_n\) as \(n\) approaches infinity must be zero, i.e., \(\lim_{n \to \infty} a_n = 0\).

If both conditions are met, the alternating series converges. However, if \(a_n\) does not approach zero, the series diverges regardless of whether the terms decrease. This test is especially useful for series like the alternating harmonic series, where the terms decrease and approach zero.

Here are two examples demonstrating the Alternating Series Test:

1. Converging Example:

Series: \(\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n + 1}}\)

Solution:

1. Decreasing terms: \(\frac{1}{\sqrt{n+1}}\) is decreasing because each term is smaller than the previous one as \(n\) increases.
2. Limit approaching zero: \(\lim_{n \to \infty} \frac{1}{\sqrt{n+1}} = 0\)

Since both conditions are satisfied, the series converges by the Alternating Series Test.

2. Diverging Example:

Series: \(\sum_{n=1}^{\infty} \frac{(-1)^n}{\ln(n)}\)

Solution:

1. Decreasing terms: \(\frac{1}{\ln(n)}\) is decreasing for \(n≥2\), as \(\ln(n)\) increases with \(n\).
2. Limit not approaching zero: \(\lim_{n \to \infty} \frac{1}{\ln(n)} = 0\), but the terms decay too slowly for the series to converge.

Since the terms decrease, but not quickly enough, the series diverges.