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- Feb 14, 2012
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Show that \(\displaystyle \frac{1}{44}>\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\left(\frac{5}{6}\right)\cdots\left( \frac{1997}{1998}\right)>\frac{1}{1999}\)
The other inequality is easily proved by induction. Let $P_n = \Bigl(\dfrac{1}{2}\Bigr)\Bigl(\dfrac{3}{4}\Bigr) \cdots \Bigl(\dfrac{2n-1}{2n}\Bigr).$ Then $P_n > \dfrac{1}{2n+1}$.Show that \(\displaystyle \frac{1}{44}>\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\left(\frac{5}{6}\right)\cdots\left( \frac{1997}{1998}\right)>\frac{1}{1999}\)