Question

The value of $\underset{x\to \infty }{lim}\left\{\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...\frac{1}{\left(n-1\right)-n}\right\}is$

Answer 1

$\underset{n\to \infty }{lim}\left(\frac{1}{1.2}+\frac{1}{2.2}+\frac{1}{2.4}+....+\frac{1}{(n-1)-n}\right)$

$\underset{n\to \infty }{lim}\left(\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+...+\frac{n-\left(n-1\right)}{\left(n-1-n\right)}\right)$

$\underset{n\to \infty }{lim}\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{n-1}-\frac{1}{n}\right)$

$\underset{n\to \infty }{lim}\left(1-\frac{1}{n}\right)$

$1asn\to \infty$