I'm trying to understand a proof in Chandrasekharan's Introduction to Analytic Number Theory. Specifically, the proof of the lemma on p.118 before Dirichlet's theorem on primes in arithmetic progressions.
Define
$ Q(s) = \log P(s) $
for some particular branch of the logarithm for $\sigma > 1$. If
$ Q(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}, $
which converges absolutely for $\sigma > 1$ and such that the coefficients $a_n$ are nonnegative, how can I conclude that
$ P(s) = e^{Q(s)} = 1 + Q(s) + \frac{Q^2(s)}{2!} + \cdots $
can be written as a Dirichlet series which is convergent for $\sigma > 1$ and whose coefficients are nonnegative?
I know that a product of Dirichlet series with nonnegative coefficients is again a Dirichlet series of nonnegative coefficients which converges on the intersection of the two half-planes of convergence, and hence each of the terms here is a Dirichlet series, but I don't see why an infinite sum of Dirichlet series is necessarily itself a Dirichlet series.
Also, how do I show that, if the Dirichlet series of $Q(s)$ converges, so does the Dirichlet series of $P(s)$, and vice versa?