Nonexplosion criterion for a pure birth process
= Nonexplosion criterion for a pure birth process
A pure birth process is nonexplosive exactly when
$$
\sum_{n=0}^{\infty}\frac1{\lambda_n}=\infty.
$$
Its explosion time is the sum of independent exponential holding times. If the displayed series converges, that sum has finite expectation. If it diverges, the Laplace transform of the sum is
$$
\prod_n\frac{\lambda_n}{\lambda_n+s}=0,
$$
forcing the sum to be infinite almost surely.