I know that I can use Lebesgue or monotone convergence theorem to exchange limit of partial sums and a Lebesgue integral, given a power series or a generic function series. But in general given a series $\sum\_{n=0}^{\infty}a\_n$ which converges, and defined $\int\_0^\infty\sum\_{n=0}^{\infty}a\_n f\_n(u)du$ with $f\_n(u)$ integrable, I was wondering when I could exchange the integration and the series. In particular in the context of Borel summation , given $\int\_0^\infty e^{-u} \sum\_{n=0}^{\infty}\frac{a\_nu^n}{n!}du$, I was wondering how could I demonstrate that if $\sum\_{n=0}^{\infty}a\_n$ converges, then I can exchange the integral and the series. (I know that for power series $\sum\_{n=0}^{\infty}a\_n z^n$ the work can be done using the radius of convergence and I can always find a dominant)



Exchanging series and integrals
As suggested by Gerald Edgar, we can use the Fubini--Tonelli theorem. By the Tonelli theorem,
$$\int\_0^\infty \sum\_{n=0}^{\infty}\Big|e^{-u} \frac{a\_nu^n}{n!}\Big|\,du
=\sum\_{n=0}^{\infty}\frac{|a\_n|}{n!}\int\_0^\infty e^{-u} u^n\,du
=\sum\_{n=0}^{\infty}|a\_n|<\infty.$$
So, the Fubini theorem is applicable, that is, one can interchange the integral and the series.


