Question

\(\lim\limits_{\rightarrow}\left(\dfrac{1}{1.3}+\dfrac{1}{2.4}+...+\dfrac{1}{n\left(n+2\right)}\right)\)

Answer 1

\(lim\left(\dfrac{1}{1.3}+\dfrac{1}{2.4}+...+\dfrac{1}{n\left(n+2\right)}\right)\)

\(=lim\left[\left(\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{\left(n-1\right)\left(n+1\right)}\right)+\left(\dfrac{1}{2.4}+\dfrac{1}{4.6}+...+\dfrac{1}{n\left(n+2\right)}\right)\right]\)

\(=lim\left(\dfrac{1}{2}\left(1-\dfrac{1}{n+1}+\dfrac{1}{2}-\dfrac{1}{n+2}\right)\right)\)

\(=lim\left(\dfrac{1}{2}.\left(\dfrac{3}{2}-\dfrac{2n+3}{n^2+3n+2}\right)\right)\)

\(=\dfrac{3}{4}\)