I’m stuck on the following probability question: Calculate the limit
\begin{aligned}\cdot \\ \lim _{n\rightarrow \infty }\sum ^{n^{2}+3n}_{k=n^{2}+2n+1}e^{-n^{2}}\dfrac{n^{2k}}{k!}\end{aligned}
One direction I thought of was to use the binomial approximation \begin{aligned}\text{bin}(n,\lambda/n) \longrightarrow\:\text{Poisson}(\lambda) \quad\text{in distribution} \end{aligned} and then use the CLT on the sum of the i.i.d. random variables that make up the binomial distribution. Problem is that this method just seems too complicated, and it seems like it’s the wrong way to go.