Convergence of the integral $ \intop_{2}^{\infty}\frac{x-\left\lfloor x\right\rfloor-\frac12}{\ln x} dx$

Question

I want to determine if the integral

$ \intop_{2}^{\infty}\frac{x-\left \lfloor{x}\right \rfloor-\frac{1}{2}}{\ln x} $ converges, and to determine if its absolute convergence or in condition.

Here's what I tried:

notice that

$ |\intop_{2}^{t}\left(x-\left\lfloor {x}\right\rfloor -\frac{1}{2}\right)dx|\leq|\intop_{2}^{\left\lfloor t\right\rfloor }\left(x-\left\lfloor {x}\right\rfloor -\frac{1}{2}\right)|+|\intop_{\left\lfloor t\right\rfloor }^{t}\left(x-\left\lfloor {x}\right\rfloor -\frac{1}{2}\right)|\leq0+1 $

and my explanation for the claim

$ |\intop_{2}^{\left\lfloor t\right\rfloor }\left(x-\left\lfloor {x}\right\rfloor -\frac{1}{2}\right)|=0 $

is this figure:

Also, $ \frac{1}{\ln x} $ is monotonic and $ \lim_{x\to\infty}\frac{1}{\ln x}=0 $.

Thus, from Dirichlet's test, the integral converges.

My questions are:

1. How to prove my point (that the red area is always 0 for the natural numbers) in an algebric formal way ?
2. How to determine if this integral is absolute convergent or just in condition ?

Thanks in advance.

Answer 1

$$-\frac{1}{8}\leq \int_{0}^{t}\{x\}-\frac{1}{2}\,dx \leq 0$$

and the absolute value of (the fraction part of $x$ minus $\frac{1}{2}$) is a continous function with a positive mean value ($\frac{1}{4}$), hence the integral is convergent by Dirichlet's test but it is not absolutely convergent by summation by parts.