= Solution
Integrating $e^{tx}$ against the <Gamma density> gives the <moment-generating function of a gamma distribution>:
$$
M_X(t)=\frac{\lambda^r}{(r-1)!}\int_0^\infty x^{r-1}e^{-(\lambda-t)x}\,dx
=\boxed{\left(\frac{\lambda}{\lambda-t}\right)^r},\qquad t<\lambda.
$$
The <change of variables> $y=(\lambda-t)x$ reduces the integral to $\Gamma(r)=(r-1)!$ using the <gamma function>. For $t\geq\lambda$ the defining integral is infinite. In particular the <moment-generating function> is finite on a <neighbourhood> of zero.
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