Tổng quát: \(\frac{2}{\left(a-1\right)a\left(a+1\right)}=\frac{1}{\left(a-1\right).a}-\frac{1}{a\left(a+1\right)}\)

Ta có: \(S=\frac{2}{1.2.3}+\frac{2}{2.3.4}+.....+\frac{2}{2013.2014.2015}\)

\(S=\left(\frac{1}{1.2}-\frac{1}{2.3}\right)+\left(\frac{1}{2.3}-\frac{1}{3.4}\right)+.....+\left(\frac{1}{2013.2014}-\frac{1}{2014.2015}\right)\)

\(S=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+....+\frac{1}{2013.2014}-\frac{1}{2014.2015}\)

\(S=\frac{1}{1.2}-\frac{1}{2014.2015}=\frac{1}{2}-\frac{1}{2014.2015}<\frac{1}{2}\)

Vậy....................

S=(2/1.2-2/2.3)+(2/2.3-2/3.4)+(2/3.4-2/4.5)+...........+(2/2013.2014-2/2014-2/2015)

S=(2/1.2-2/2014.2015):2

S=1-2/2014.2/2015

--> S>1/2

\(S=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{2013.2014.2015}\)

\(S=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{2013.2014}-\frac{1}{2014.2015}\right)\)

\(S=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2014.2015}\right)\)

\(S=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4058210}\right)\)

\(S=\frac{1}{2}.\left(\frac{2029105}{4058210}-\frac{1}{4058210}\right)\)

\(S=\frac{1}{2}.\frac{2029104}{4058210}\)

\(S=\frac{1014552}{4058210}\)

Chúc bạn học tốt !!! 

Công thức : 

\(\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}\right)=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{6}\right)=\frac{1}{2}.\left(\frac{3}{6}-\frac{1}{6}\right)=\frac{1}{2}.\frac{2}{6}=\frac{1}{6}=\frac{1}{1.2.3}\)

Ta có: \(\frac{3n+2}{n\left(n+1\right)\left(n+2\right)}\)

\(=\frac{2n+2+n}{n\left(n+1\right)\left(n+2\right)}=\frac{2\left(n+1\right)}{n\left(n+1\right)\left(n+2\right)}+\frac{n}{n\left(n+1\right)\left(n+2\right)}\)

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

\(=\frac{1}{n}+\frac{1}{n+1}-\frac{2}{n+2}\)

Do đó, ta có: \(\frac{5}{1\cdot2\cdot3}=\frac{3\cdot1+2}{1\cdot2\cdot3}=\frac11+\frac{1}{1+1}-\frac{2}{1+2}=1+\frac12-\frac23\)

\(\frac{8}{2\cdot3\cdot4}=\frac{3\cdot2+2}{2\cdot3\cdot4}=\frac12+\frac13-\frac24\)

...

Do đó, ta có: \(S=1+\frac12-\frac23+\frac12+\frac13-\frac24+\frac13+\frac14-\frac25+\ldots+\frac{1}{n}+\frac{1}{n+1}-\frac{2}{n+2}\)

\(=1+\left(\frac12+\frac12\right)+\left(-\frac23+\frac13+\frac13\right)+\left(-\frac24+\frac14+\frac14\right)+\cdots+\left(-\frac{2}{n}+\frac{1}{n}+\frac{1}{n}\right)-\frac{2}{n+1}+\frac{1}{n+1}-\frac{2}{n+2}\)

\(=1+1-\frac{1}{n+1}-\frac{2}{n+2}<2\)

=>\(S_{2022}=\frac{5}{1\cdot2\cdot3}+\frac{8}{2\cdot3\cdot4}+\cdots+\frac{6068}{2022\cdot2023\cdot2024}<2\)

Ta có: M=\(\frac{1}{1.2.3}\) +\(\frac{1}{2.3.4}\) +\(\frac{1}{3.4.5}\) +...+\(\frac{1}{100.101.102}\) 

         M=2.(\(\frac{1}{1.2.3}\) +\(\frac{1}{2.3.4}\) +\(\frac{1}{3.4.5}\) +...+\(\frac{1}{100.101.102}\) ).\(\frac{1}{2}\)

          M=(\(\frac{2}{1.2.3}\) +\(\frac{2}{2.3.4}\) +\(\frac{2}{3.4.5}\) +...+\(\frac{2}{100.101.102}\) ).\(\frac{1}{2}\)

          M=(\(\frac{1}{1.2}\) -\(\frac{1}{2.3}\) +\(\frac{1}{2.3}\) -\(\frac{1}{3.4}\) +\(\frac{1}{3.4}\) -\(\frac{1}{4.5}+...+\frac{1}{100.101}-\frac{1}{101.102}\) ).\(\frac{1}{2}\)

          M=( \(\frac{1}{1.2}-\frac{1}{101.102}\)).\(\frac{1}{2}\)

          Mà \(\frac{1}{1.2}-\frac{1}{101.102}<1\)

         Và \(\frac{1}{2}<1\) 

        \(=>\)  (\(\frac{1}{1.2}-\frac{1}{101.102}\) ) .\(\frac{1}{2}\) \(<1\)

        \(=>\) M <1

A = 1 - 1/2 - 1/3 + 1/2 - 1/3 - 1/4 + ... + 1/2014 - 1/2015 - 1/2016

A = 1- 1/2016

A = 2015/2016

A > 1/4

A=4949/19800 và 1/4

4949/19800 < 1/4

Xong! t*** mik đê!!!

\(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{2014.2015.2016}\)

\(A=\frac{1}{2}\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{2015.2016}\right)\)

\(A=0,2499998...<0,25\)

\(\Rightarrow A<\frac{1}{4}\)

bạn viết gì vậy? Mình ko thấy?