Answer
$n\gt 1.07\times 10^{11301}$
Hence, it is proved we need more than $10^{11301}$ terms
Work Step by Step
$R_{n}\leq \int_{n}^{\infty}f(x) dx \lt 0.000000005$
$\lim\limits_{a \to \infty} \int_{n}^{a}x^{-1.001} dx\lt 0.000000005$
$\lim\limits_{a \to \infty}[-\frac{1}{-0.001}x^{-0.001}]_{n}^{a}\lt 0.000000005$
$\frac{1000}{n^{0.001}}\lt 0.000000005$
$n^{0.001}\gt \frac{1000}{0.000000005}$
$n\gt 1.07\times 10^{11301}$
Hence, it is proved we need more than $10^{11301}$ terms
