Extension: Similarities of Infinite Series and Improper Integrals

The harmonic series is $$\sum^\infty_{i=1}\frac{1}{i}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots$$ This happens to be a right Riemann sum of the function $\frac{1}{x-1}$. Does $\frac{1}{x-1}$ have finite or infinite area from 1 to $\infty$?