Let $\ X:=X_{k\,n}\ $ be a random variable of a $n$-sided die where $\Pr(X=i)=\frac{1}{n}$ for each $i\in\{1,2,\ldots,n\},\ $ where $\ k\in\{2, \ldots,n-1\}\ $ and $\ n\ $ are fixed. Let $t$ be a parameter that varies according to the outcome of each roll. Initially $t=1$ and we stop rolling the die when $t=0$.

Each time when the value $x$ taken by $X$ is less than or equal to $\ k\ $ we win $x$ dollars and we decrease $t$ by $1$. Each time when $x$ 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:= k(n)\ $ 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 $0$?