We have $$e^{2\pi i n}=1$$
So we have $$e^{2\pi in+1}=e$$
which implies $$(e^{2\pi in+1})^{2\pi in+1}=e^{2\pi in+1}=e$$
Thus we have $$e^{-4\pi^{2}n^{2}+4\pi in+1}=e$$
This implies $$e^{-4\pi^{2}n^{2}}=1$$
Taking the limit when $n\rightarrow \infty$ gives $0=1$.
Answer
Your error is (as in most of those fake-proofs) in the step where you use the power law $(a^b)^c=a^{bc}$ without the conditions of that power law being fulfilled.
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