I wanted to add that there is a reason WHY we like to formalize things very strictly.
In the 19th century mathematicians started studying things like trigonometric series and Fourier series and such, and there were a lot of questions about what exactly it did mean for a series to converge, and what was the difference between regular and uniform convergence.
Historically, mistakes were made. Someone would think they proved convergence but they were wrong. It gradually became clear that they needed to have a precise formal definition of convergence. So this didn’t happen in a vacuum. There was a practical need to have clear definitions because things were getting messy.
So 19th century math started with the same mindset that you have. We “know” what the sum of an infinite series is and we can work with them intuitively. It only became clear gradually, over time, that a formal theory was needed. That’s why the 20th century was all about formalization. Not for the sake of being formal for its own sake; but to avoid errors caused by intuitive and imprecise thinking.