Let $X$ be the random variable of a $n$-sided die where $\Pr(X=i)=\frac{1}{n}$ for each $i\in\{1,2,\ldots,n\}$. Let $t$ be the number of times we have to roll the die before we stop, which can change according to the outcomes. Initially $t=1$ and we stop rolling the die when $t=0$. 

Each time the value $x$ taken by $X$ is less than or equal to a given value $k\in\{2, 3 \ldots, n-1\}$, we win $x$ dollars and we decrease $t$ by $1$. Each time $x$ is larger than $k$, we lose $x$ dollars and we increase $t$ by adding $x-1$ to its current value. We have an unlimited amount of money.

What is the minimum value of $k$ such that the expected number of dollars obtained (i.e., the difference between the total number of dollars won and the total number of dollars lost) is larger than $0$?