Question

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

Answer 1

$1+\frac{1}{1+2}+\frac{1}{1+2+3}+\dots \frac{1}{1+2+\dots n}$
$=\frac{2}{1\cdot 2}+\frac{2}{2\cdot 3}+\frac{2}{3\cdot 4}+\dots \frac{2}{n(n+1)}$
$=2(\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\dots \frac{1}{n(n+1)})$
$=2(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\cdots \frac{1}{n}-\frac{1}{n+1})$
$=2(1-\frac{1}{n+1})$
For infinite terms, $n\to \infty$ gives sum as: $2$