Sum of series $\sum_{n=1}^{\infty}\frac{1}{n(n+\frac{1}{2})}$ $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n(n+\frac{1}{2})}$ i am trying to solve an integral and this ended up being last part I need, Wolfram alpha gives $4-2\ln{4}$ but I don't know how they actually got that value. can anyone help or give a starting point? telescoping series didn't work

Hint: $a_n = \dfrac{4}{2n} - \dfrac{4}{2n + 1}$.
