Answer
Diverges
Work Step by Step
Let us consider that $I_n=\int_{1}^{n} \dfrac{10}{x} \ dx$
After integrating, we have: $I_n=10 [\ln |x|]_1^{n} =10 \ln |n|$
Now, $I=\lim\limits_{n \to \infty} I_n=\int_{1}^{\infty} \dfrac{10}{x} \ dx\\=\lim\limits_{n \to \infty} 10 \ln |n|\\=\infty$
This means that the limit does not exist and the given integral diverges.