How to calculate the limit $\lim_{n\rightarrow\infty} n\int^{1}_{0}x^{kn}e^{x^{n}}dx$ Let $k$ be a fixed positive integer. How to calculate the following limit? $$\lim_{n\rightarrow\infty} n\int^{1}_{0}x^{kn}e^{x^{n}}dx$$

The change of variable $t=x^n$ yields
$$
n\int^{1}_{0}x^{kn}\mathrm e^{x^{n}}\mathrm dx=\int^{1}_{0}t^{k-1}\mathrm e^tt^{1/n}\mathrm dt,
$$
and $t^{1/n}\to1$ when $n\to\infty$ hence the LHS converges to
$$
I_k=\int^{1}_{0}t^{k-1}\mathrm e^t\mathrm dt.
$$
A standard recursion on $k\geqslant1$ then leads to
$$
I_k=\mathrm e\,(k-1)!\sum_{i=0}^{+\infty}\frac{(-1)^i}{(k+i)!}.
$$
Sanity checks: $I_1=\mathrm e-1$ and $I_k\to0$ when $k\to\infty$.
