so sanh A va B A=(1/2 ² -1).(1/3 ² -1).(1/4 ²-1)…….(1/2014 ²-1) B= -1/2

Question

so sanh A va B
A=(1/2 ² -1).(1/3 ² -1).(1/4 ²-1)…….(1/2014 ²-1)
B= -1/2

in progress 0
2 tháng 2021-10-05T17:05:12+00:00 2 Answers 7 views 0

1. $A = (\dfrac{-3}{2^2}) (\dfrac{-8}{3^3})…(\dfrac{1-2014^2}{2014^2})$

$A= -(\dfrac{3}{2.2} \dfrac{8}{3.3} … \dfrac{4056195}{2014^2})$

(Do co (2014-2):1+1 = 2013 so am nen co dau am o dau).

$A=-(\dfrac{1.3}{2.2} \dfrac{2.4}{3.3}…\dfrac{2013.2015}{2014^2})$

$A = -(\dfrac{1.2.3….2013}{2.3.4….2014} \dfrac{3.4.5…2015}{2.3.4….2014}$

$A = -(\dfrac{1}{2014} \dfrac{2015}{2}) = \dfrac{-2015}{4028}.$

Lai co $B = -1/2 = -2014/4028$

Vay \$A

2. $\begin{array}{l} A = \left( {\frac{1}{{{2^2}}} – 1} \right)\left( {\frac{1}{{{3^2}}} – 1} \right)\left( {\frac{1}{{{4^2}}} – 1} \right)……\left( {\frac{1}{{{{2014}^2}}} – 1} \right)\\ Trong\,\,\,bieu\,\,thuc\,\,A\,\,co\,\,tat\,ca:\,\,\,\frac{{2014 – 2}}{2} + 1 = 1007\,\,phan\,\,tu.\\ \Rightarrow A\,\,co\,\,gia\,\,tri\,\,am\,\,hay\,\,A < 0.\\ \Rightarrow A = \left( {\frac{1}{2} - 1} \right)\left( {\frac{1}{2} + 1} \right)\left( {\frac{1}{3} - 1} \right)\left( {\frac{1}{3} + 1} \right)\left( {\frac{1}{4} - 1} \right)\left( {\frac{1}{4} + 1} \right).......\left( {\frac{1}{{2014}} - 1} \right)\left( {\frac{1}{{2014}} + 1} \right)\\ = - \left[ {\left( {1 - \frac{1}{2}} \right)\left( {1 - \frac{1}{3}} \right)\left( {1 - \frac{1}{4}} \right)......\left( {1 - \frac{1}{{2014}}} \right)} \right]\left[ {\left( {1 + \frac{1}{2}} \right)\left( {1 + \frac{1}{3}} \right)\left( {1 + \frac{1}{4}} \right)......\left( {1 + \frac{1}{{2014}}} \right)} \right]\\ = - \left( {\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.....\frac{{2013}}{{2014}}} \right).\left( {\frac{3}{2}.\frac{4}{3}.\frac{5}{4}.....\frac{{2015}}{{2014}}} \right)\\ = - \frac{1}{{2014}}.\frac{{2015}}{2} = - \frac{1}{2}.\frac{{2015}}{{2014}} < - \frac{1}{2}.\\ \Rightarrow A < B. \end{array}$