a: \(T_n=\dfrac{1}{2}\left(\dfrac{1}{1\cdot2}-\dfrac{1}{2\cdot3}+\dfrac{1}{2\cdot3}-\dfrac{1}{3\cdot4}+...+\dfrac{1}{n\left(n+1\right)}-\dfrac{1}{\left(n+1\right)\left(n+2\right)}\right)\)
=>\(T=\dfrac{1}{2}\cdot\left(\dfrac{1}{2}-\dfrac{1}{\left(n+1\right)\left(n+2\right)}\right)\)
=>\(T=\dfrac{1}{2}\cdot\dfrac{n^2+3n+2-2}{2\left(n+1\right)\left(n+2\right)}=\dfrac{n^2+3n}{4\left(n+1\right)\left(n+2\right)}\)
b: \(S_n=1+\dfrac{1}{8}+\dfrac{1}{27}+\dfrac{1}{4^3}+...+\dfrac{1}{n^3}\)
=>\(S< \dfrac{251}{216}+\dfrac{1}{2}\left(\dfrac{1}{3\cdot4}-\dfrac{1}{4\cdot5}+\dfrac{1}{4\cdot5}+...+\dfrac{1}{\left(n-1\right)\cdot n}-\dfrac{1}{n\left(n+1\right)}\right)\)
=>\(S< \dfrac{251}{216}+\dfrac{1}{24}-\dfrac{1}{2n\left(n+1\right)}\)
=>\(S< \dfrac{260}{216}-\dfrac{1}{2n\left(n+1\right)}< \dfrac{65}{54}\)
