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Math Help - series of sign function

  1. #1
    MHF Contributor Amer's Avatar
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    series of sign function

    let x_1,x_2,...,x_n be real numbers
    define

    y_n = sign\left(\prod_{i=1}^{n} x_i \right)

    such that

    sign(x) = \begin{cases} \;1 & \;\;  x>0 \\ \;0 &\;\; x=0 \\ -1 & \;\; x<0 \end{cases}

    I tried to write \sum_{i=1}^{n}y_i this what i get

    \sum_{i=1}^{n}y_i = \begin{cases}<br />
n &\;\;, x_i >0 \;\;\;\;\;\; \forall i\in \{1,...,n\} \\ <br />
n-1 & \;\;, x_i>0 \;\;\;\;\;\; \forall i\in \{1,...,n-1\},x_n=0  \\ <br />
n-2 &\;\;,x_i>0 \;\; \forall i\in\{1,...,n\}/\{j,k\} ,\;\text{such that}\;\;x_j,x_k<0\;and\;\mid j-k\mid =1 \\ <br />
n-2 &\;\;,x_i>0\;\; \forall i\in\{1,...,n-2\},x_{n-1}=0 \\ <br />
... & \\ <br />
 -n+1&\;\;,x_1<0\;\;\;\;,and\;\;\;x_j>0, \;\;\forall\;\; j\in \{2,...,n-1\},x_n=0  \\ <br />
-n &\;\;, x_1 <0 \;\;\;\;,and\;\;\;x_j>0, \;\;\forall\;\; j\in \{2,...,n\} <br />
\end{cases}

    it is clear that the series diverge my questions are is my work correct, dose this series has a name ??, and what is the <br/> in my latex the last code I copied it from online latex equation editor

    Thanks
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  2. #2
    Member
    Joined
    Jun 2010
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    Israel
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    Re: series of sign function

    Quote Originally Posted by Amer View Post
    let x_1,x_2,...,x_n be real numbers
    define

    y_n = sign\left(\prod_{i=1}^{n} x_i \right)

    such that

    sign(x) = \begin{cases} \;1 & \;\;  x>0 \\ \;0 &\;\; x=0 \\ -1 & \;\; x<0 \end{cases}

    I tried to write \sum_{i=1}^{n}y_i this what i get

    \sum_{i=1}^{n}y_i = \begin{cases}<br />
n &\;\;, x_i >0 \;\;\;\;\;\; \forall i\in \{1,...,n\} \\ <br />
n-1 & \;\;, x_i>0 \;\;\;\;\;\; \forall i\in \{1,...,n-1\},x_n=0  \\ <br />
n-2 &\;\;,x_i>0 \;\; \forall i\in\{1,...,n\}/\{j,k\} ,\;\text{such that}\;\;x_j,x_k<0\;and\;\mid j-k\mid =1 \\ <br />
n-2 &\;\;,x_i>0\;\; \forall i\in\{1,...,n-2\},x_{n-1}=0 \\ <br />
... & \\ <br />
 -n+1&\;\;,x_1<0\;\;\;\;,and\;\;\;x_j>0, \;\;\forall\;\; j\in \{2,...,n-1\},x_n=0  \\ <br />
-n &\;\;, x_1 <0 \;\;\;\;,and\;\;\;x_j>0, \;\;\forall\;\; j\in \{2,...,n\} <br />
\end{cases}

    it is clear that the series diverge my questions are is my work correct, dose this series has a name ??, and what is the <br/> in my latex the last code I copied it from online latex equation editor

    Thanks
    I don't know whether it helps...
    A simple observation: Among the y_i, if there are t positive numbers and s negative numbers, then \sum_{i=1}^{n}y_i=t-s.
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  3. #3
    Super Member ILikeSerena's Avatar
    Joined
    Dec 2011
    Posts
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    Re: series of sign function

    Quote Originally Posted by Amer View Post
    , and what is the <br/> in my latex the last code I copied it from online latex equation editor
    Interesting.

    Apparently every new line in a TEX expression is converted into a HTML line break.
    This is not supposed to be the way it works.

    The only way I see around it, is not to use line breaks.
    So either a giant one liner, or separate TEX lines.
    (Or otherwise an admin should fix the TEX engine.)
    Thanks from Amer
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