# Math Help - series of sign function

1. ## 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 <br/> in my latex the last code I copied it from online latex equation editor

Thanks

2. ## Re: series of sign function

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 <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 $\sum_{i=1}^{n}y_i=t-s$.

3. ## Re: series of sign function

Originally Posted by Amer
, 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.)