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.