What's the significance of conditional and absolute convergence?

Question

I know how to find if a series is conditional or absolute convergence by taking the absolute value and seeing if that absolute sum converges (then the original sum is absolute). If the absolute diverges but the original converges, then that's conditional.

But my question is: is there a significance? Is absolute convergence series bigger than absolute? Does a series being absolute or conditional affect rate of convergence, or radius? Does it affect symmetry?

Looking at Google, I only found tests on how to find if it's absolute or conditional. But I want to find meaning or significance. Or at least more mathematical properties.

Answer 1

Let $ \Sigma x_n$ be a series of real numbers. Now consider splitting $x_n$ into two separate series: $p_n$, and $q_n$, where: $p_n = \begin{cases}\ \ x_n, & x_n > 0 \\ 0 & \text{otherwise} \end{cases}$ and $p_n = \begin{cases}\ \ -x_n, & x_n < 0 \\ 0 & \text{otherwise} \end{cases}$

It can easily be shown that

1. $ \Sigma x_n$ = $ \Sigma p_n - \Sigma q_n$
2. $ \Sigma |x_n|$ = $ \Sigma p_n + \Sigma q_n$

What happens if $x_n$ is absolutely convergent? Well this must imply that $ \Sigma p_n$ and $ \Sigma q_n$ are convergent. Now, more interestingly, what if $ \Sigma x_n$ is conditionally convergent? Then $ \Sigma p_n$ and $ \Sigma q_n$ must both diverge by 2. Therefore $ \Sigma x_n$ is a difference of divergent series. Heuristically, this means that the two series are somehow "taming" each other.

Another interesting consequence of this is the Riemman rearrangement theorem that uses this fact to rearrange any conditionally convergent sequence so that it converges to any real number.

So absolute convergence is somehow more "stable" than absolute convergence.

Answer 2

There's a theorem that proves that if a series is conditionally convergent then it is possible to rearrange the terms of the sum so that it converges to any value. Basically, if you sum the positive and negative terms separately you'll get $\infty - \infty$, an indeterminate form that depends on how the two terms are diverging. The bottom line is that an absolutely convergent series is more robust to manipulations without having to worry about whether the answer is altered by them.

Mathologer has a good video on this aspect of conditional convergence.