Question

The sum of infinity of the series $\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}$ .......... is equal to .......... .

Answer 1

Denominator of $n^{th}$ term is sum of first $n$ natural numbers i.e $\frac{\left(n\right)(n+1)}{2}$
$n^{th}$ term of equation
$T_n=\frac{1}{\frac{n(n+1)}{2}}=\frac{2}{n(n+1)}=\frac{2}{n}-\frac{2}{n+1}$
Sum of $n$ terms
$S_n=\sum T_n=\frac{2}{1}-\frac{2}{2}+\frac{2}{2}-\frac{2}{3}..........+\frac{2}{n}-\frac{2}{n+1}$
$\Rightarrow S_n=2-\frac{2}{n+1}\Rightarrow S_{\infty }=2$