Reduce the integral For each nonnegative integer $n$, there exists a polynomial $p_n(x)$ such that
$$\int x^ne^xdx=p_n(x)e^x+C$$
Let
$$L=\lim_{n \to \infty}\frac{|p_n(2)|}{n!}$$
Find $⌊1000L⌋$.

Let $$I_{n}=\int x^ne^xdx=x^ne^x-n\int e^xx^{n-1}dx=x^ne^x-nI_{n-1}$$
so
$$I_{n}=x^ne^x-n(x^{n-1}e^x-(n-1)I_{n-2})=e^xx^n-nx^{n-1}e^x+n(n-1)x^{n-2}e^x-n(n-1)(n-2)I_{n-3}=\cdots$$
we have
$$I_{n}=e^{x}\sum_{i=0}^{n}(-1)^i\dfrac{n!}{(n-i)!}x^{n-i}=e^x\sum_{i=0}^{n}(-1)^{n-i}\dfrac{n!}{i!}x^i$$
so
$$\dfrac{|P_{n}(2)|}{n!}=\sum_{i=0}^{n}(-1)^{i}\dfrac{2^i}{i!}$$
so
$$\lim_{n\to\infty}\dfrac{|P_{n}(2)|}{n!}=\sum_{i=0}^{\infty}(-1)^{i}\dfrac{2^i}{i!}=e^{-2}$$
so
$$[1000L]=[1000e^{-2}]=135$$
