Is the result of this infinite division correct? [duplicate]
Solution 1:
All of these and similar "tricks" for evaluating infinite expressions are really arguments of the form,
"If the expression converges to some number $x$, then the following must be true about $x$..."
So you have indeed proven that if your expression converges, it does so to either $i$ or $-i$, and not to some other value. But you haven't proven that the expression converges in the first place (and it is easy to see that it does not).