1. ## Improper integral

Hi,
Determen whether the integral of(x-[x]-0.5)/ln(x) from 2 to infinity converges,absolutely converges or diverges,
where [x] denotes the lower integer part of x

I tried to write the integral as the infinite sum of integrals from n to n+1 of (x-n-0.5)/ln(x) but so far with no specific consequences.
also, the numerator in each interval [n,n+1] is symmetric around x=n+0.5 so translation may help.
I need some help
Thank's again

HI,
Determen whether the integral of(x-[x]-0.5)/ln(x) from 2 to infinity converges,absolutely converges or diverges,
where [x] denotes the lower integer part of x

I tried to write the integral as the infinite sum of integrals from n to n+1 of (x-n-0.5)/ln(x) but so far with no specific consequences.
also, the numerator in each interval [n,n+1] is symmetric around x=n+0.5 so translation may help.
I need some help
Thank's again

2. ## Re: Improper integral

For the sum of integrals I got $\displaystyle \sum_{n=2}^\infty\int_n^{n+1}\left(\frac{x}{\ln x} - \frac{n}{\ln x} - \frac{1}{2\ln x}\right) dx$.

The denominator breaks the symmetry such that the absolute value at the upper bound will be smaller than that at the lower bound.

Spoilers: https://www.wolframalpha.com/input/?...rate+x%2F(lnx) , https://www.wolframalpha.com/input/?...rate+1%2F(lnx)

Is the original function even integrable in the range $\displaystyle [2, \infty)$? It has an infinite number of discontinuities in that range (a function bounded on a closed interval and with a finite number of discontinuities on that interval is integrable). I suspect breaking it up as that sum is valid, but I'm not a mathematician.