= Solution
The displayed density is that of a <gamma distribution> with shape $k+1$ and rate one. Put
$$
K=\sum_{i=1}^nk_i.
$$
By the <additivity of independent gamma distributions with a common rate>,
$$
X_1+\cdots+X_n\sim\operatorname{Gamma}(K+n,1).
$$
Equivalently, the <independence of random variables> makes the <Laplace transform of a nonnegative random variable> of the sum equal
$$
\prod_{i=1}^n(1+s)^{-(k_i+1)}
=(1+s)^{-(K+n)}.
$$
Therefore the required <probability density function> is
$$
\boxed{
f_{X_1+\cdots+X_n}(x)
=\mathbf1_{\{x>0\}}
\frac{x^{K+n-1}e^{-x}}{(K+n-1)!}.}
$$
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