Evaluate $\sum_{n=2}^{17} \frac{n^{2}+n+1}{n^{4}+2 n^{3}-n^{2}-2 n}$.
Solution: Observe that the denominator $n^{4}+2 n^{3}-n^{2}-2 n=n(n-1)(n+1)(n+2)$. Thus we can rewrite the fraction as $\frac{n^{2}-n+1}{n^{4}+2 n^{3}-n^{2}-2 n}=\frac{a}{n-1}+\frac{b}{n}+\frac{c}{n+1}+\frac{d}{n+2}$ for some real numbers $a, b, c$, and $d$. This method is called partial fractions. Condensing the right hand side as a fraction over $n^{4}+2 n^{3}-n^{2}-2 n$ we get $n^{2}-n+1=a\left(n^{3}+3 n^{2}+2 n\right)+b\left(n^{3}+2 n^{2}-n-2\right)+c\left(n^{3}+n^{2}-2 n\right)+d\left(n^{3}-n\right)$. Comparing coefficients of each power of $n$ we get $a+b+c+d=0,3 a+2 b+c=2,2 a-b-2 c-d=2$, and $-2 b=2$. This is a system of 4 equations in 4 variables, and its solution is $a=1 / 2, b=-1 / 2, c=1 / 2$, and $d=$ $-1 / 2$. Thus the summation becomes $\frac{1}{2}\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots+\frac{1}{16}-\frac{1}{17}+\frac{1}{18}-\frac{1}{19}\right)$. Notice that almost everything cancels to leave us with $\frac{1}{2}\left(1+\frac{1}{3}-\frac{1}{17}-\frac{1}{19}\right)=\frac{\mathbf{5 9 2}}{\mathbf{9 6 9}}$.