Approximate distribution

Question

I'm doing some work in random graphs and there is some probability case. I have some approximation which turns out to be with density: $\mathbb{P}[X = k] = {\lambda^k \over k! }\cdot e^{-\lambda} \cdot \left( 1 + O\left( {1\over n} \right) \right)$ for some $n$. Could we find some properties of this distribution? (for example $X \sim \text{Pois}\left(\lambda \cdot \left( 1 + O\left( {1\over n} \right)\right) \right)$ or something like this? I'm not talking about some limit distribution, $1 + O\left( {1\over n} \right)$ can be read by something, close to 1.