Question

Find the sum of n terms of the series:
$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+......$

Answer 1

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

$T_n=\frac{1}{n\times (n+1)}=\frac{(n+1)-\left(n\right)}{\left(n\right)(n+1)}=\frac{1}{n}-\frac{1}{n+1}$

$T_1=\frac{1}{1}-\frac{1}{2}$

$T_2=\frac{1}{2}-\frac{1}{3}$

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$T_n=\frac{1}{n}-\frac{1}{n+1}$

$T_1+T_2+T_3+....T_n=S=1-\frac{1}{n+1}$

$S=\frac{n}{n+1}$