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 $p_n \rightarrow 0$.
1 Answer
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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.