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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.

user3556214
  • 88
  • What do your symbols mean? In particular $b_{nk}$ and the ordered pair. – 9301293 Nov 13 '15 at 20:16
  • $b_{nk}$ is a real valued sequence with indices $n$ and $k$.

    $a_{jnk}(x,h)$ does only mean that it is a real valued sequence which depends on the indices $j,n,k \in \mathbb{N}$ and the real numbers $x,h$.

     – user3556214 Nov 14 '15 at 01:47

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