Question

$\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+.........$
Sum to simplify ?

Answer 1

Let $S=\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+.....\text{upto n terms}$

General term $=\frac{1}{\left(\frac{n(n+1)}{2}\right)}=\frac{2}{n(n+1)}=2(\frac{1}{n}-\frac{1}{n+1})$

Therefore,

$S=2(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.....+\frac{1}{n-1}-\frac{1}{n}+\frac{1}{n}-\frac{1}{n+1})$

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

$\Rightarrow S=2\left(\frac{n}{n+1}\right)=\frac{2n}{n+1}$