For define . Prove that . So consider an interval . Let . Now we want to show that . Then we take and conclude that on . The second series is convergent.
But no, g(s) is not necessarily equal to 0. I'm not too sure (I hate those limit/convergence theorems), but proving that may be enough (because both terms of the difference converge (*))
Consider the integral
Then divide the integral on intervals which boundaries are integers :
But this is a telescoping sum !
So we have :
And this obviously goes to 0 as N goes to infinity (because )
and this converges
According to the defintion reported in...
Floor Function -- from Wolfram MathWorld
... the 'fractional part of x' is defined as...
... and it is the function you see in figure...
In the interval is...
According to (1) and remembering that...
... we obtain...
... so that...
From (5) it derives that for ...