The Alternating Series Test (AST) is a helpful way to figure out if a series converges when the signs of the terms switch back and forth.

What is an Alternating Series?

An alternating series looks like this:

$\sum_{n=1}^{\infty} (-1)^{n} a_n$

Here, $a_n$ is a group of positive numbers.

Conditions for the AST:

For the series to converge (which means it approaches a specific value), two rules must be followed:

1. The terms must always get smaller. This means that $a_{n+1}$ should be less than or equal to $a_n$ for all larger values of $n$.

2. The terms must get closer and closer to zero. Specifically, we need $\lim_{n \to \infty} a_n = 0$.

These rules make it easy to test if the series converges without having to figure out the sum.

Conditional vs. Absolute Convergence:

It’s also important to know the difference between conditional and absolute convergence.

• A series is conditionally convergent if it passes the AST test but the series of absolute values ($\sum_{n=1}^{\infty} a_n$) doesn’t converge.

• On the other hand, if the series of absolute values ($\sum_{n=1}^{\infty} |(-1)^n a_n|$) does converge, we say the series is absolutely convergent.

To sum up:

• The AST gives clear rules to check if alternating series converge.

• It shows how important it is to list conditions for testing convergence.

• Knowing the difference between conditional and absolute convergence helps us better understand series convergence in calculus.