# How do you evaluate the sum represented by sum_(n=1)^(8)1/(n+1) ?

Oct 22, 2014

Begin by changing the denominators to $1 + 1 , 2 + 1 , 3 + 1$ and so on to $8 + 1. . .$

Next add $\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9}$

This requires a common denominator. If we multiply $9 \cdot 8 \cdot 7 \cdot 5$ we will get $2520$.

The $9$ picks up multiples of the $3$, the and the $8$ picks up multiples of the $2 , 3 \mathmr{and} 4$.

Now, multiply $\frac{1}{2} \cdot \frac{1260}{1260}$ giving $\frac{1260}{2520}$. Multiply $\frac{1}{3} \cdot \frac{840}{840}$ giving $\frac{840}{2520}$.

$\frac{1}{4} \cdot \frac{630}{630} = \frac{630}{2520} , \frac{1}{5} \cdot \frac{504}{504} = \frac{504}{2520} , \frac{1}{6} \cdot \frac{420}{420} = \frac{420}{2520} , \frac{1}{7} \cdot \frac{360}{360} = \frac{360}{2520} , \frac{1}{8} \cdot \frac{315}{315} = \frac{315}{2520} \mathmr{and} \frac{1}{9} \cdot \frac{280}{280} = \frac{280}{2520.}$

$\frac{1260 + 840 + 630 + 504 + 420 + 360 + 315 + 280}{2520}$ which = $\frac{4609}{2520}$.