Determine whether $\sum_{n=1}^\infty (-1)^n \Big(1-\frac{2}{n}\Big)^n$ converges or diverges $$\sum_{n=1}^\infty (-1)^n \Big(1-\frac{2}{n}\Big)^n$$
If these terms were all positive I would have no problem concluding that the series diverges since the limit would be $e^{-2}>0$. However, I don't understand how the divergence test works with $(-1)^n$ added in. Is this still divergent? Both the Ratio and Root tests are inconclusive, and Wolfram Alpha is scratching its head. When I put partial sums above 1000, it spits out imaginary numbers somehow. Yet it seems to stay around 1.

If it's convergent then $\left | a_{n} \right |\rightarrow 0$
$(1-\frac{2}{n})^{n}=((1+\frac{1}{-2^{-1}n})^{-2^{-1}n})^{2}\rightarrow e^{2}\neq 0$
and therefore isn't convergent.
