Show that $\int_{1}^{\infty}\frac{x-\left \lfloor x \right \rfloor -\frac{1}{2}}{x^{s+1}}(1-s\log x)\,\mathrm{d}x$ is convergent for $\Re s > -1$

Question

I am struggling to understand the example in Special Functions p. 621, that states, that $$\int_{1}^{\infty}\frac{x-\left \lfloor x \right \rfloor -\frac{1}{2}}{x^{s+1}}(1-s\log x)\,\mathrm{d}x$$ is convergent for $\Re s > -1$ by Abel's test. I do not see how it is deduced using the Abel's test when it is about the infinite series. Is it possible to show it the other way?

Answer 1

Dirichlet's test is enough: $\{x\}-\frac{1}{2}$ is a function with a bounded primitive and $\frac{1-s\log x}{x^{s+1}}$ is monotonically convergent to zero from some point on.