Answer
$1000$
Work Step by Step
Consider $f(x)=\int_{1}^{\infty} \dfrac{1}{x^{1.001}} dx$
$\lim\limits_{a \to \infty} f(x)= \lim\limits_{a \to \infty}\int_{1}^{a}\dfrac{1}{x^{1.001}} dx=\lim\limits_{a \to \infty} [\dfrac{-x^{1.001}}{0.001}]_{1}^{a}$
or, $\lim\limits_{a \to \infty} [-1000a^{-0.001}+1000]= 1000$
