# Find sum of first n terms of series : $\;1+\large\frac{1}{1+2}+\large\frac{1}{1+2+3}+\;....$

$(a)\;\large\frac{n}{n-1}\qquad(b)\;\large\frac{2n}{n+1}\qquad(c)\;\large\frac{2n}{n-1}\qquad(d)\;\large\frac{n}{n+1}$

Answer : (b) $\;\large\frac{2n}{n+1}$
Explanation : $a_{r}=\large\frac{1}{\large\frac{r(r+1)}{2}}$
$=\large\frac{2}{r(r+1)}$
$=2\;[\large\frac{1}{r}-\large\frac{1}{r+1}]$
$S=\sum_{r=1}^{n}\;a_{r}$
$=\sum_{r=1}^{n}\;2\;[\large\frac{1}{r}-\large\frac{1}{r+1}]$
$=2\;[1-\large\frac{1}{n+1}]$
$=\large\frac{2n}{n+1}.$
answered Jan 23, 2014 by 1 flag