It will probably help if you realise that since there's a common denominator, you can write

1/n^2 + 2/n^2 + 3/n^2 + ... + (n - 1)/n^2 = [1 + 2 + 3 + ... + (n-1)]/n^2 = [n(n-1)/2]/n^2 = (n^2 - n)/2n^2 = 1/2 - 1/2n.

I'm sure you can take the limit now.