総乗の極限問題

\begin{align*} \lim_{n\rightarrow\infty}\prod_{k=1}^{n}\left(1+\frac{k}{n^{2}}\right) & =\lim_{n\rightarrow\infty}\prod_{k=1}^{n}\left(1+\frac{k}{n^{2}}\right)^{\frac{n^{2}}{k}\cdot\frac{k}{n^{2}}}\\ & =\lim_{n\rightarrow\infty}\prod_{k=1}^{n}e^{\frac{k}{n^{2}}}\\ & =\lim_{n\rightarrow\infty}\prod_{k=1}^{n}\pow\left(e^{\frac{1}{n^{2}}},k\right)\\ & =\lim_{n\rightarrow\infty}\pow\left(e^{\frac{1}{n^{2}}},\sum_{k=1}^{n}k\right)\\ & =\lim_{n\rightarrow\infty}\pow\left(e^{\frac{1}{n^{2}}},\frac{n\left(n+1\right)}{2}\right)\\ & =\lim_{n\rightarrow\infty}e^{\frac{n+1}{2n}}\\ & =\sqrt{e} \end{align*}