Here some interesting theorem about your firs issue:
Riemann series theorem - Wikipedia, the free encyclopedia
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.
Here some interesting theorem about your firs issue:
Riemann series theorem - Wikipedia, the free encyclopedia
I am not sure if it is essential whether we are talking about a series or just about a limit of a numerical sequence. Yes, there are different definitions of real numbers, but one of them says that a real number is an equivalence class of converging sequences of rational numbers. Basically, a real number is not a single "point": it's a function , or, more precisely, a whole family of such functions.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?
Real numbers are not really physical. For example, a real number can store an infinite amount of information. So it makes sense to view them as abstractions such as limits.