TITLE: What's wrong with this reasoning that $\frac{\infty}{\infty}=0$?
QUESTION [61 upvotes]: $$\frac{n}{\infty} + \frac{n}{\infty} +\dots = \frac{\infty}{\infty}$$

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 $0$.

REPLY [5 votes]: You cannot use an "infinite" distributive law with division by the symbol $\infty$. In other words the statement
$$\frac{\sum_{n=1}^\infty a_n}{\infty} = \sum_{n=1}^\infty \frac{a_n}{\infty}$$
is not valid if $\sum_n a_n$ is a divergent series. For the left-hand side cannot have any meaning when that sequence diverges, whereas the right-hand side is the zero series (provided that each $a_n$ is an ordinary finite number) which has the sum $0$ for any reasonable summation definition.
The law $\frac{a+b}{d}=\frac{a}{d}+\frac{b}{d}$ is true for real or complex numbers $a$, $b$ and $d$ provided $d\ne 0$, but it does not quite generalize to the form you use.