My book asks to simplify this problem:
$$\sum_{i=1}^n\frac{i}{n^2}$$
This equals:
$$\sum_{i=1}^n\frac{1}{n^2}i$$
Now, shouldn't
$$\sum_{i=1}^n\frac{1}{n^2}=\frac{1}{n}\text{ ?}$$
Because you are summing $1/n^2$, $n$ times, so that $n / n^2 = 1/n$.
So it should be
$$\frac{1}{n}\sum_1^n i = \frac{1}{n} \cdot \frac{n(n+1)}{2} = \mathbf{\frac{n+1}{2}}$$
But the answer in my book is $\mathbf{(n+1)/2n}$.