Solution
= Solution
The <partial fraction decomposition>
$$
f(i)=\frac1{i(i+1)}=\frac1i-\frac1{i+1}
$$
makes the sum a <telescoping sum>. All interior terms cancel:
$$
\boxed{\sum_{i=1}^nf(i)=\left(1-\frac12\right)+\left(\frac12-\frac13\right)+\cdots+\left(\frac1n-\frac1{n+1}\right)=1-\frac1{n+1}=\frac n{n+1}.}
$$