级数绝对收敛与函数绝对可积性的差异及深层原因问询

Hey folks, I’ve been banging my head against this intuitive disconnect between series convergence and integral convergence, and I’m hoping someone can help me see the deeper picture here.

Here’s what I’ve got so far:

• Using the Cauchy criterion, it’s super straightforward to show that if $\sum |f_n|$ converges, then $\sum f_n$ must also converge. I follow the proof, but when I shift over to integrals, the logic breaks down.
• For integrals, just because $\int |f|$ exists (meaning $|f|$ is integrable) doesn’t guarantee that $\int f$ exists. A perfect example is the modified Dirichlet function: $f(x) = 1$ if $x \in \mathbb{Q}$ and $-1$ otherwise. The integral of $|f|$ is just the integral of 1 over the interval, which exists, but $\int f$ never converges—since the upper and lower sums can never meet.

I get the formal proofs for both cases, but I’m curious if there’s a deeper, more intuitive reason why these two "sum-like" operations behave so differently here. After all, integrals are often described as generalized sums, so why does the Cauchy criterion work seamlessly for series but doesn’t translate cleanly when we take the limit of partition sizes going to zero for integrals?

备注：内容来源于stack exchange，提问作者Frido