How would you write this set of linear programming problems as latex?

Question

I am not able to make self understood over at Mathematics Stackexchange in this question.

Therefore I ask here the Mathematica experts how to write these linear programming problems as latex, the standard way of communicating mathematics at the stack exchange network:

(*start*)
nn = 18;
TableForm[
  L2 = Table[
    LinearProgramming[
     Table[1/n, {n, 1, k}], {Table[If[n == 1, k, 1], {n, 1, k}]}, {{1,
        0}}, Table[
      If[n == 1, {-1, 1}, {-2 (n - 1), 0 (n - 1)}], {n, 1, k}]], {k, 
     1, nn}]]
(*end*)

Here is my own try:

$$\begin{array}{ll} \text{minimize} & \displaystyle\sum_{n=1}^{n=k} \frac{x_{n}}{n} \\ \text{subject to constraints:} & k + \displaystyle\sum_{n=2}^{n=k}x_{n}=1 \\ & x_1 \geq -1 \end{array}$$

for all $k$ and for $n>1$ $$-2(n-1) \leq x_n \leq 0 \tag{4}$$