I know how to evaluate limits and infinite series and have a basic understanding of them I guess. I also know the epsilon-delta definition of a limit. However, I am having a bit of trouble seeing how infinite series really work.

I've seen that .999...repeating is equal to 1 and that they are essentially the same. I've seen the method in representing .999...repeating as an infinite series which converges to 1 which convinces me they are the same thing.

Say that some infinite series converges to some number n. Does this mean that the infinite series approaches and converges to n but never reaches n or does it mean that this number n can be defined as this infinite series, or something else?

Another issue is the area of the Koch Curve

Koch snowflake - Wikipedia, the free encyclopedia

Which initially made me think about this. The area is said to be finite; it's 8/5ths the area of the zeroth iteration. The area is derived from using a convergent infinite series to sum up all the small triangles. Does this mean that the area is approaching this value but never quite reaches it or is the area definitely 8/5ths the original area?

I believe these ideas are developed more rigorously in Mathematical Analysis, but I have no knowledge in Analysis.

Thank you.