I am currently the convergence of different processes. Doing this, I ended up with this expression and was wondering whether it is true that$$\lim\_{n \rightarrow \infty}\sum\_{k=0}^{n} \frac{|(1-\frac{n p\_n}{n})|^{n-k}- e^{- \lambda}|}{k!}=0?$$ -For the case that $np\_n \rightarrow \lambda $ and $p\_n \rightarrow 0$.



Is $\lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{n p_n}{n})|^{n-k}- e^{- \lambda}|}{k!}=0$?
I'd say it's true, by the dominated convergence theorem for series. You may think the $n$-th sum as a series whose terms $c\_{n,k} $ vanish for $k > n$. For fixed $k$, the coefficient $c\_{n,k}$ tends to $0$ as $n\to\infty$ (recall that if $z\_n\to z$ then $(1+z\_n/n)^n\to e^z$). Moreover $|c\_{n,k}|$ is uniformly bounded in $n$ by $C^k/k!$ so that the series for any $n$ are actually dominated by a convergent series.


