What's wrong with this reasoning that $\frac{\infty}{\infty}=0$? 
$$\frac{n}{\infty} + \frac{n}{\infty} +\dots = \frac{\infty}{\infty}$$

You can always break up $\infty/\infty$ into the left hand side, where n is an arbitrary number. However, on the left hand side $\frac{n}{\infty}$ is always equal to $0$. Thus $\frac{\infty}{\infty}$ should always equal $0$.

When you learned how to extend arithmetic from the natural numbers to the integers to the rational numbers to the real numbers and to the complex numbers, one of the main motivations was to preserve the laws of arithmetic.
The situation with the extended real numbers (i.e. the real numbers along with $\pm \infty$) is different — the goal is not to preserve the laws of arithmetic, the goal is to have continuity.
Consequently, arguments that involve naively applying the ordinary laws arithmetic to extended real numbers are quite unreliable.
This is compounded by the fact the argument involves infinite summation. Infinite sums can fail to satisfy many of the laws that finite sums obey, so that's a second source of unreliability.
