I am reading the paper "Limit distributions for multitype branching processes of m-ary search trees" by Chauvin, Liu and Pouyanne. In section 5, the following random variable is defined:
$Z=X_1+\dots+X_n,$
where the $X_i$ are $\text{Exp}(i)$ distributed and independent of each other. It says that the density of $Z$ is
$f_Z(u)=n\cdot e^{-u} \cdot (1-e^{-u})^{n-1}$ for $u>0$ and $0$ else.
How do I compute this density, because it says that it should be easy to see, but for now I haven't found a fast and easy way. I found a formula for independent exponentially distributed random variables with different parameters and wanted to do it with induction, but this turns out to be a quite long computation.
