Learning about series of functions can be really tough for students, mostly because of the tricky idea of convergence. The terms pointwise convergence and uniform convergence can feel a bit confusing. Without a clear understanding of these ideas, students often have a hard time using them correctly.

First, let’s break down what pointwise convergence means. It means that for every single point in the domain, the series converges at that point. But figuring out how this can change at different points can be puzzling. For example, take the series $\sum_{n=1}^{\infty} f_n(x)$. How it acts at different $x$ values might not feel very clear.

Now, let’s look at uniform convergence. This type of convergence has stricter rules. The series converges uniformly if the speed of convergence is the same, no matter which point in the domain you pick. This difference can be easily missed, which causes confusion. Students might wonder, "Why should I care about uniform convergence?" The answer is important because it affects how we can switch between limits and integrals, which can change the answers we get.

Students also face some tricky definitions that require a good understanding of epsilon-delta language. Ideas like $\epsilon$-neighborhoods and limit points can sound scary and confusing, making students feel lost.

Finally, the need for strong proof techniques can be scary. Figuring out counterexamples or conditions for convergence can feel like a complicated puzzle, which can discourage many learners. To really get these concepts, students need to practice and be willing to tackle problems, but this might stop some from even trying.