Using the Comparison Test for Divergence

Show that the series \(\sum\limits_{k=1}^{\infty }\dfrac{k+3}{k(k+2)}\) diverges.

Solution Since \(\dfrac{k+3}{k(k+2)}\) has a factor \(\dfrac{1}{k}\), we choose to compare the given series to the harmonic series \(\sum\limits_{k=1}^{\infty }\dfrac{1}{k}\), which diverges. \[ \frac{n+3}{n(n+2)}=\left( {\frac{n+3}{n+2}}\right) \left( {\frac{1}{n}} \right) >\frac{1}{n} \]

It follows from the Comparison Test for Divergence that \(\sum\limits_{k=1}^{\infty }\dfrac{k+3}{k(k+2) }\) diverges.