i have a question about asymptotic equivalence which means
$ \lim_{n \to \infty} \frac{f(n)}{g(n)}=1$ with notation $f(n) \sim g(n)$.
I know that the following holds:
$\sum_{j=0}^{\infty} a_{jnk}(h) \sim \int_{0}^{\infty}a_{nk}(x,h)dx \hspace{0,1cm} \forall $ fixed $ n,k \in \mathbb{N} $ and for $ h \to \infty $
My question is now if the following implication holds or under which conditions it holds?
$\sum_{j=0}^{\infty} \sum_{n=1}^{\infty} \sum_{k=1}^{\infty} a_{jnk}(h) b_{nk} \sim \int _{0}^{\infty} \sum_{n=1}^{\infty} \sum_{k=1}^{\infty} a_{nk}(x,h) b_{nk} dx$ for $ h \to \infty$
Thanks for help.