My instinct is to use the root test, and try to show thatProve that

$\displaystyle \sum\left(1-\frac{1}{n}\right)^{n^2}$

converges.

$\displaystyle \lim_{n\to\infty}\left(1-\frac{1}{n}\right)^n<1$.

But unfortunately I don't know how to show this either.

Are there any tricks I can use on these nth root proofs which might help here?

Thanks!