Question

Find the sum of $n$ terms. Also find the sum of infiite terms
$1+\frac{1}{1+2}+\frac{1}{1+2+3}+...$

Answer 1

Given series is,

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

$⟹{\sum }_{r=1}^r\frac{2}{r(r+1)}$

$⟹{\sum }_{r=1}^r\frac{2\left[\right(r+1)-r]}{r(r+1)}$

$⟹{\sum }_{r=1}^{r}2[\frac{1}{r}-\frac{1}{r+1}]$

On, putting values, we get,

$2[\frac{1}{1}-\frac{1}{2}]$+$2[\frac{1}{2}-\frac{1}{3}]$+$2[\frac{1}{3}-\frac{1}{4}]$+...……+$2[\frac{1}{n}-\frac{1}{n+1}]$

$⟹2[\frac{1}{1}-\frac{1}{n+1}]$ ………………………………(1)

$⟹\frac{2n}{n+1}$

For sum of infinite series , put $r\to \infty$ in (1), we get

$⟹2$