series of sign function

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December 24th 2011, 11:51 AM
Amer

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

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

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

Thanks
December 24th 2011, 12:49 PM
melese

Re: series of sign function

Quote:

Originally Posted by Amer

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

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

it is clear that the series diverge my questions are is my work correct, dose this series has a name ??, and what is the 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 $\sum_{i=1}^{n}y_i=t-s$ .
December 24th 2011, 01:15 PM
ILikeSerena

Re: series of sign function

Quote:

Originally Posted by Amer

, and what is the 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.)