#### Answer

$\displaystyle \frac{617}{140}$

#### Work Step by Step

$\displaystyle \sum_{k=1}^{6}\frac{k}{k+1}=$
$=\displaystyle \frac{1}{1+1}+\frac{2}{2+1}+\frac{3}{3+1}+\frac{4}{4+1}+\frac{5}{5+1}+\frac{6}{6+1}$
$=\displaystyle \frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}+\frac{6}{7}$
$=\displaystyle \frac{210+280+315+336+350+360}{420}$
$=\displaystyle \frac{1851}{420}$
$=\displaystyle \frac{617}{140}$