Question Number 104092 by naka3546 last updated on 19/Jul/20

Prove that (1/2) ∙ (3/4) ∙ (5/6) ∙ …∙ ((2005)/(2006)) ∙ ((2007)/(2008)) < (1/(√(2009)))

Commented byJDamian last updated on 19/Jul/20

I guess ((2007)/(2009)) should actually be ((2007)/(2008))

Answered by 1549442205PVT last updated on 19/Jul/20

It is easy to see that if (a/b)<1 then a<b ⇒ab+a<ab+b⇒a(b+1)<b(a+1)⇒(a/b)<((a+1)/(b+1)). Hence, Putting A=(1/2).(3/4).(5/6)....((2005)/(2006)).((2007)/(2009)).We have: (1/2)<(2/3),(3/4)<(4/5),(5/6)<(6/7)...,((2005)/(2006))<((2006)/(2007)),((2007)/(2009))<((2009)/(2010)) Multiplying 1004 inequlities side by side we getA<(2/3).(4/5).(6/7)....((2006)/(2007)).((2009)/(2010))=B ⇒A^2 <AB=(1/2).(2/3).(3/4).(4/5).(5/6)....((2005)/(2006)).((2006)/(2007)).((2007)/(2009)).((2009)/(2010)) =(1/(2010))⇒A<(1/(√(2010)))<(1/(√(2009)))(q.e.d)