Question

Prove that:

$1+\frac{1}{1+2}+\frac{1}{1+2+3}+.....\frac{1}{1+2+3+....n}=\frac{2n}{n+1}$.

Answer 1

Let the $rth$ term is $T_r$

$\Rightarrow T_r=\frac{1}{1+2+3\cdots +r}$

Therefore, sum of the series can be written as

$\Rightarrow \sum _{r=1}^n\frac{2}{r(r+1)}$

Add and subtract $2r$ in numerator, we get

$\sum _{r=1}^n\frac{2(r+1)-2r}{r(r+1)}=2\sum _{r=1}^n\frac{1}{r}-\frac{1}{r+1}$

$\Rightarrow 2[1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}\dots \cdots +\frac{1}{n}-\frac{1}{n+1}]$

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