stolz+夹逼/洛必达：
$$=\lim_{n\rightarrow\infty}\frac{\frac{e^{n+1}}{n+1}}{\int_{n}^{n+1}e^{x^2}dx}=\lim_{n\rightarrow\infty}\frac{e^{n+1}}{\int_{n}^{n+1}(n+1)e^{x^2}dx}$$又有：
$$\int_{n}^{n+1}xe^{x^2}dx<\int_{n}^{n+1}(n+1)e^{x^2}dx<\int_{n}^{n+1}xe^{x^2}dx+\int_{n}^{n+1}e^{x^2}dx$$$$左边\int_{n}^{n+1}xe^{x^2}dx=\frac{1}{2}(e^{(n+1)^2}-e^{n^2})$$$$右边\int_{n}^{n+1}xe^{x^2}dx+\int_{n}^{n+1}e^{x^2}dx = \frac{1}{2}(e^{(n+1)^2}-e^{n^2})+e^{\xi^2}\qquad\xi < n+1$$于是由夹逼准则：
$$\lim_{n\rightarrow\infty}\frac{\sum^n_{k=1}\frac{e^{k^2}}{k}}{\int_0^ne^{x^2}dx}=2$$