Question

Show that, 1+$\frac{1}{\left(1+2\right)}+\dots +\frac{1}{\left(1+2+3\dots n\right)}=\frac{2n}{\left(n+1\right)}$

Answer 1

Given series is $1+\frac{1}{1+2}+\frac{1}{1+2+3}+............\frac{1}{1+2+\mathrm{3....}n}$

$T_1=1=\frac{2}{1}-\frac{2}{2}$

$T_2=\frac{1}{3}=\frac{2}{2}-\frac{2}{3}$

$T_n=\frac{2}{n(n+1)}=\frac{2}{n}-\frac{2}{n+1}$

on on surrounding up all things

$S_n=T_1+T_2+......T_n$

$S_n=(\frac{2}{1}-\frac{2}{2})+(\frac{2}{2}-\frac{2}{3})+....(\frac{2}{n}-\frac{2}{n+1})$

$S_n=2-\frac{2}{n+1}=\frac{2n}{n+1}$