Question

1 Can we do the infinite series summation ?
Can we do this in one minute inside the mind (without putting a pen on paper?

2 What is the N th term ?
3 What is the sum up to N terms ?

$S=\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+....$

Answer 1

Infinite series.

Reciprocal of each term is an AP series sum.

$T_n=\frac{2}{n(n+1)}=2*[\frac{1}{n}-\frac{1}{n+1}]\\\\S_N=2*\Sigma_{n=1}^N\ [\frac{1}{n}-\frac{1}{n+1}]\\\\=2*[\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-....]\\\\=2$