Is $\psi(x)-\log x$ strictly increasing for strictly positive $x$?

Question

Let $\psi(x)$ be the digamma function. Is the function which takes $\psi(x)-\log x$ for $x>0$ strictly increasing, and how could one show this if it is the case (link etc.)?

Answer 1

Are you allowed to use $\psi(z) = -\gamma + \sum_{n=0}^\infty \left(\frac{1}{n + 1} - \frac{1}{n + z}\right)$ ? If so $$\psi'(x)-\frac1x = \sum_{n\ge 0} \frac1{(x+n)^2}-\int_{x+n}^{x+n+1} \frac1{t^2}dt$$