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Math Help - Notion of Limits and Infinite Series

  1. #1
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    Notion of Limits and Infinite Series

    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.
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Here some interesting theorem about your firs issue:

    Riemann series theorem - Wikipedia, the free encyclopedia
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  3. #3
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    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?
    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 \mathbb{N}\to\mathbb{Q}, or, more precisely, a whole family of such functions.

    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.
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