Finite sum

Evaluate the sum:

$$S= \frac{1}{1+2}+ \frac{1}{1+2+3}+\cdots + \frac{1}{1+2+3+\cdots +2015}$$

Solution



The denominators are successive terms of an arithmetic progression. Hence:

$${\begin{aligned}
S &=\frac{1}{\frac{2\cdot 3}{2}}+ \frac{1}{\frac{3\cdot 4}{2}}+ \cdots + \frac{1}{\frac{2015 \cdot 2016}{2}} \\
 &= \frac{2}{2\cdot 3+ 3\cdot 4+\cdots+ 2015\cdot 2016}\\
 &= \left (\frac{2}{2}\bcancel{-\frac{2}{3}}  \right )+ \left ( \bcancel{\frac{2}{3}}- \frac{2}{4} \right )+\cdots +\left ( \bcancel{\frac{2}{2015}}- \frac{2}{2016} \right )\\
 &= 1- \frac{2}{2016}\\
 &=\frac{1007}{1008}
\end{aligned}}$$