Answer
Diverges
Work Step by Step
Let us consider that $I_n=\int_{10}^{n} (\dfrac{1}{x}-10) \ dx$
After integrating, we have: $I_n=[\ln |x|-10x]_{10}^{n} = \ln |\dfrac{n}{10}|-10(n-10)$
Now, $I=\lim\limits_{n \to \infty} I_n=\int_{10}^{\infty} (\dfrac{1}{x}-10) \ dx\\=\lim\limits_{n \to \infty} [\ln |\dfrac{n}{10}|-10(n-10)] \\=\infty$
Because the logarithm and $n$ both are unbounded. So, the limit does not exist. This means that the given integral diverges.