Asymptotically isometric copy of $\ell_1$

Question

A Banach space $X$ is said to contain Asymptotically isometric copy of $\ell_1$ if there is a null sequence$(\rho_n)_{n=1}^{\infty}$ in $(0,1)$ and a sequence$(x_n)_{n=1}^{\infty}$ in $X$ so that $$\sum\limits_{n=1}^{\infty}(1-\rho_n)|a_n|\leq\left\|\sum\limits_{n=1}^{\infty}a_nx_n\right\|\leq\sum\limits_{n=1}^{\infty}|a_n|,$$ for all $(a_n)_{n=1}^{\infty}\in\ell_1.$ I really do not know very well this definition, especially geometry interpretation of this definition, I would appreciate any help.