Why is the force on the charge at the tip of a cone infinite? Imagine a charge $q$ that is located at the top of a hollow cone with a surface charge density $\sigma.$ The slant height is $L$ and the charge $q$ sits at the vertex of angle $2\theta$. We are interested in the force acting on the charge $q$.

Assume that there are shells inside this hollow cone. Let $l$ be some slant height. If you took a ring with radius $r$ and thickness $dl$ the surface area is $2\pi(l\sin\theta) dl$. The forces in the horizontal cancel out, so we're left with the vertical force. The charge $dQ$ is $\sigma 2\pi (l \sin\theta) dl$ 
$$dF = \dfrac{qdQ\cos\theta}{4\pi \epsilon l^2} = \dfrac{q(\sigma 2\pi l\sin\theta dl)\cos\theta}{4\pi \epsilon l^2},$$ but when you integrate over the interval $[0, L]$ you get:
$$ F= \dfrac{q\sigma \sin\theta \cos\theta}{2\epsilon} \int_0^{L} \dfrac{dl}{l} $$ which doesn't converge. I was wondering experimentally has this ever been verified before? 
 A: The general prescription for  force between two charges is 
$F =  \frac{k \; q \; Q }{r^2}$
What happens when the distance between the charges is $0$? If we remove the integration over the cone that is essentially the problem you are stating. For instance if you place the charge a little more higher than the top of the cone then your integral will give a finite value.
So we need to look for answer to the question, "can you really bring two finite charges to coincide over each other in space so that $r=0$" Remember that charges at micro scale have carriers in the form of electrons, protons etc and there are also Quantum laws to contend with. 
I think it is simply  not possible to place a charge $q$ at the charged tip of a cone and make both the charges to remain static within their respective structures, and then finally to experimentally measure their force of interaction. 
So what will happen when you try to do this as an experiment. If the charged cone is bolted stationary, and lets say you a bring a very small object of charge $q$ along the vertical axis and you keep measuring the force as you approach the charged tip. I think just before you reach the tip of the cone the electric field would be large enough that charges will flow and discharge as spark, till they are at the same potential. (Vacuum arc if the experiment was done in vacuum) 
A: Potential at vertex of cone, letting zero at infinite, is
$$V= \frac{\Phi}{2\epsilon}\cdot\frac{RH}{L}$$
Where
$\Phi=$ surface charge density
$\epsilon=$ permittivity of that space
$L=$ slant length $=\sqrt{R^2+H^2}$
