\(A=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)\left(\frac{1}{4^2}-1\right)\cdot\cdot\cdot\cdot\left(\frac{1}{2013^2}-1\right)\left(\frac{1}{2014^2}-1\right)\)

\(A=\left(\frac{-3}{4}\right)\left(\frac{-8}{9}\right)\left(\frac{-15}{16}\right)\cdot\cdot\cdot\left(\frac{-4052168}{4052169}\right)\left(\frac{-4056195}{4056196}\right)\)

\(A=\frac{-1\cdot3}{2\cdot2}\cdot\frac{-2\cdot4}{3\cdot3}\cdot\frac{-3\cdot5}{4\cdot4}\cdot....\cdot\frac{-2012\cdot2014}{2013\cdot2013}\cdot\frac{-2013\cdot2015}{2014\cdot2014}\)

\(A=\frac{-1\cdot\left(-2\right)\cdot\left(-3\right)\cdot....\cdot\left(-2012\right)\cdot\left(-2013\right)}{2\cdot3\cdot4\cdot....\cdot2013\cdot2014}\cdot\frac{3\cdot4\cdot5\cdot....\cdot2014\cdot2015}{2\cdot3\cdot4\cdot....\cdot2013\cdot2014}\)

\(A=\frac{-1}{2014}\cdot\frac{2015}{2}=\frac{-2015}{4028}\)

Ta thấy \(\frac{-2015}{4028}< \frac{-1}{2}\) \(\Rightarrow A< B\)

Ta có

\(A=\frac{\left(1^2-2^2\right)\left(1^2-3^2\right).....\left(1^2-2014^2\right)}{\left(2.3.4.....2014\right)\left(2.3....2014\right)}\)

\(\Leftrightarrow A=\frac{\left(-1\right)3\left(-2\right)4.....\left(-2013\right)2015}{\left(2.3.4.....2014\right)\left(2.3....2014\right)}\)

\(\Leftrightarrow A=\frac{\left[\left(-1\right)\left(-2\right)...\left(-2013\right)\right]\left(3.4.5...2015\right)}{\left(2.3.4.....2014\right)\left(2.3....2014\right)}\)

\(\Leftrightarrow A=\frac{\left(-1\right)2015}{2014.2}=-\frac{2015}{4028}< -\frac{2014}{4028}=-\frac{1}{2}\)

=> A<-1/2

A có: \(\frac{2014-2}{3-2}+1=2013\) ( thừa số )

Ta thấy mỗi thừa số của A đều có dạng \(\frac{1}{n^2}-1\)với \(n\inℕ^∗\)và \(n>1\)

Có \(\frac{1}{n^2}< 1\Rightarrow\frac{1}{n^2}-1< 1-1=0\)

=> Mỗi thừa số của A đều nhỏ hơn 0

=> A là tích của 2013 thừa số nhỏ hơn 0

Mà 2013 là số lẻ

=> A < 0

Mà B = \(\frac{1}{2}\)> 0

=> A < B

\(A=\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}...\frac{2013^2-1}{2013^2}.\frac{2014^2-1}{2014^2}\)

\(A=\frac{1.3.2.4.3.5....2012.2014.2013.2015}{2^2.3^2.4^2...2013^2.2014^2}\)

\(A=\frac{\left(1.2.3...2012.2013\right).\left(3.4.5...2014.2015\right)}{\left(2.3.4...2013.2014\right).\left(2.3.4...2013.2014\right)}\)(nhóm từng số ở trước và sau vào 2 nhóm khác nhau)

\(A=\frac{3.2015}{2014.2}\)

\(A=\frac{6045}{4028}\)

\(A=\frac{6045}{4028}\),nha bạn ,chúc bạn hok tốt ,love bạn nhìu ,cách làm giống như Monozono Nanami nha

Ta có : \(\frac{1}{n^2}-1=\frac{1-n^2}{n^2}=\frac{\left(1-n\right)\left(1+1\right)}{n^2}\)

Áp dụng :

\(A=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)\left(\frac{1}{4^2}-1\right)...\left(\frac{1}{2014^2}-1\right)\)

\(=\frac{-1.3}{2.2}.\frac{-2.4}{3.3}.\frac{-3.5}{4.4}.....\frac{-2013.2015}{2014.2014}\)

\(=\frac{-\left(1.2.3...2013\right)\left(3.4.5....2015\right)}{\left(2.3.4.....2014\right)\left(2.3.4......2014\right)}=\frac{-2015}{2014.2}=\frac{-2015}{4028}\)

Sr còn thiếu

\(A=-\frac{2015}{4028}< \frac{-2014}{4028}=-\frac{1}{2}\)

Vậy \(A< B\)

\(y=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)....\left(\frac{1}{2014^2}-1\right)\)

\(y=\left(\frac{-1.3}{2.2}\right)\left(\frac{-2.4}{3.3}\right)....\left(\frac{-2013.2015}{2014.2014}\right)\)

\(y=-\left(\frac{1.2....2013.3.4...2015}{2.3....2014.2.3....2014}\right)\)

\(y=-\left(\frac{2015}{2014.2}\right)\)

\(y=\frac{-2015}{4028}\)

\(x=\frac{-1}{2}=\frac{-2014}{4028}\)

Vì \(\frac{-2015}{4028}

Vì \(\frac{1}{2^2}>0\)

 ............

 \(\frac{1}{2014^2}>0\)

=> A = \(\left(\frac{1}{2^2}\right)\left(\frac{1}{3^2}\right)...\left(\frac{1}{2014^2}\right)>0\)

B = \(-\frac{1}{2}