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Question Number 106378 by mr W last updated on 04/Aug/20

Answered by Dwaipayan Shikari last updated on 05/Aug/20

$$(1-\frac{1}{2})(1-\frac{1}{4})(1-\frac{1}{6})....n$$$$\underset{n=1}{\prod ^n}(1-\frac{1}{2n})=y$$$$log\left(\underset{n=1}{\prod ^n}(1-\frac{1}{2n})\right)=logy$$$$\underset{n=1}{\sum ^n}log(1-\frac{1}{2n})=logy$$$$\frac{\underset{n=1}{\sum ^n}log(1-\frac{1}{2n})}{n}⩾{\left(\underset{n=1}{\prod ^n}log(1-\frac{1}{2n})\right)}^{\frac{1}{n}}$$$$\frac{log(\frac{1}{2}.\frac{3}{4}......\frac{2n-1}{2n})}{n}⩾{\left(\prod ^{n}log(1-\frac{1}{2n})\right)}^{\frac{1}{n}}$$

Commented by mr W last updated on 05/Aug/20

$$thanks,butit{doesn}^{\prime }tanswerthe$$$$actualquestion.$$

Answered by mr W last updated on 05/Aug/20

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