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Thread: The definition of the limit

  1. #1
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    The definition of the limit

    Hey guys, we have started not long ago to learn the term of limes.

    So the known definition of the limit of a series goes like that:

    If
    converges to 0, then , there exists an s.t.

    I have some elementary questions about the definition, would it be correct if I use the definition a little bit? such that:

    * If s.t. we get

    * If there exists s.t. we get

    And another type of limes question:

    If the series is defined by and converges to 0 then ?

    Thank you!
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  2. #2
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    Re: The definition of the limit

    Quote Originally Posted by CStudent View Post
    Hey guys, we have started not long ago to learn the term of limes.

    So the known definition of the limit of a series goes like that:

    If
    converges to 0, then , there exists an s.t.

    I have some elementary questions about the definition, would it be correct if I use the definition a little bit? such that:

    * If s.t. we get

    * If there exists s.t. we get

    And another type of limes question:

    If the series is defined by and converges to 0 then ?
    If s.t. we get

    Look you must start with Suppose that $C>0$. Everything in the proof depends upon that positive number.
    The expression $|L-a_n|$ is the distance from $a_n$ to $L$.
    So expression $|L-a_n|<C$ means that is the distance from $a_n$ to $L$ is less than $C$.
    Now $|L-a_n|<C$ is the open interval $(L-C,L+C)$ so that $a_n\in(L-C,L+C)$.
    This also means that you can find a positive integer $N_1$ (it depends upon $C$) having the property that picking any integer, $j$, greater than $N_1$ then $a_j\in(L-C,L+C)$ or $|a_j-L|<C$

    So roughly speaking all that means is Given a positive error, C, we can find a place in the sequence where from there on the terms are close to L.
    Thanks from CStudent
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  3. #3
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    Re: The definition of the limit

    Quote Originally Posted by Plato View Post
    If s.t. we get

    Look you must start with Suppose that $C>0$. Everything in the proof depends upon that positive number.
    The expression $|L-a_n|$ is the distance from $a_n$ to $L$.
    So expression $|L-a_n|<C$ means that is the distance from $a_n$ to $L$ is less than $C$.
    Now $|L-a_n|<C$ is the open interval $(L-C,L+C)$ so that $a_n\in(L-C,L+C)$.
    This also means that you can find a positive integer $N_1$ (it depends upon $C$) having the property that picking any integer, $j$, greater than $N_1$ then $a_j\in(L-C,L+C)$ or $|a_j-L|<C$

    So roughly speaking all that means is Given a positive error, C, we can find a place in the sequence where from there on the terms are close to L.
    Great explanation, thank you!
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  4. #4
    Forum Admin topsquark's Avatar
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    Re: The definition of the limit

    Am I the only one who can't see the equations? All I'm seeing is a bunch of CodeCogs icons saying "equation quota exceeded."

    -Dan
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  5. #5
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    Re: The definition of the limit

    Dan, you are not the only one who does not see the equation. I see the same thing as you do.
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  6. #6
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    Re: The definition of the limit

    Quote Originally Posted by topsquark View Post
    Am I the only one who can't see the equations? All I'm seeing is a bunch of CodeCogs icons saying "equation quota exceeded."
    -Dan
    Try to reply with quote. I see the code at first, but then after replying I see the mathematics.
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  7. #7
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    Re: The definition of the limit

    How do I fix it?
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  8. #8
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    Re: The definition of the limit

    Quote Originally Posted by CStudent View Post
    How do I fix it?
    Replace $[\text{tex}]$ or $[\text{math}]$ tags with $\$$'s?
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  9. #9
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    Re: The definition of the limit

    Quote Originally Posted by SlipEternal View Post
    Replace $[\text{tex}]$ or $[\text{math}]$ tags with $\$$'s?
    It's just like MathJax?
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  10. #10
    Forum Admin topsquark's Avatar
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    Re: The definition of the limit

    Quote Originally Posted by CStudent View Post
    It's just like MathJax?
    Yup.

    -Dan
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