Question

thực hiện phép tính : (1-1/2^2)(1-1/3^2)(1-1/4^2)..........(1-1/2008^2)

Answer 1

\(\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right).....\left(1-\dfrac{1}{2008^2}\right)\)

\(=\dfrac{3}{4}.\dfrac{8}{9}.\dfrac{15}{16}....\dfrac{2008^2-1}{2008^2}\)

\(=\dfrac{1.3}{4}.\dfrac{2.4}{9}.\dfrac{3.5}{16}....\dfrac{2007.2009}{2008^2}\)

\(=\left(\dfrac{1.2.3...2007}{2.3.4....2008}\right).\dfrac{3.4.5...2009}{2.3.4...2008}\)

\(=\dfrac{1}{2008}.\dfrac{2009}{2}=\dfrac{2009}{4016}\)