Denote $n\_i=n\_{i-1}-\sqrt[k]{n\_{i-1}}$. If $n\_0=n$ then what is the minimum $i$ at which $n\_i<2$ holds? Is there a standard technique to solve such problems? Any references?



When does this recurrence stop?
I always like to answer these questions by approximating the difference equation with a differential equation (for which the result is always easier). In this case, the d.e. is $\dot x=-x^{1/k}$, $x(0)=n$, giving $t\approx \frac{k}{k-1}n^{(k-1)/k}$.


