6 mins ago, by Rudi_Birnbaum
OK: this one point in the 1-point compactification is it topologically equivalent to a point in a compact set?
I discovered the following relation for arbitrary $d_r$:
$$ \lim_{k \to \infty} \lim_{n \to \infty}\ \sum_{r=1}^n d_r \left( f(\frac{k}{n}r)\frac{k}{n} \right) = \lim_{s \to 1} \underbrace{\frac{1}{\zeta(s)} \sum_{r=1}^\infty \frac{d_r}{r^s}}_{\text{removable singularity}} \times \int_0^\...
First, notice that the function is defined for all reals except for $x=0$, and the points where the denominator is zero: $$\sin(1/x)=0 \iff 1/x= k\pi \iff x = \frac{1}{k\pi}$$
for integer $k$.
As you correctly guessed, that the function is not defined at $x=0$ does not matter for the limit.
B...
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations.
The sum is calculated by dividing the region up into shapes (rectangles, trapezoids, parabolas, or cubics) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together...
