Under a condition, $frac{1}{b } = sum_{n=1}^{infty}frac{1}{a_{n}}$ will never happen

Conjecture:

There is no $b,{a_n}_{n=1}^{infty}$ such
that $b,a_n in mathbb{N}^+, a_{n+1}ge a_n,lim_{nrightarrow infty}frac{a_n}{a_{n+1}}=0$ and
$$frac{1}{b}= sum_{n=1}^{infty}frac{1}{a_{n}}$$
This is just my guess, and it would be nice if someone could give a counter example.