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Question Number 119750 by 675480065 last updated on 26/Oct/20

$$\underset{\mathrm{k}=1}{\prod ^{1019}}\left[\frac{2\mathrm{k}}{2\mathrm{k}-1}\right]=?$$

Answered by Bird last updated on 26/Oct/20

$$\prod _{k=1}^{1019}\left[\frac{2k}{2k-1}\right]=\prod _{k=1}^{1019}\left[\frac{2k-1+1}{2k-1}\right]$$$$=\prod _{k=1}^{1019}[1+\frac{1}{2k-1}]$$$$=\prod _{k=1}^{1019}(1+\left[\frac{1}{2k-1}\right])$$$$=2\prod _{k=2}^{1019}(1+\left[\frac{1}{2k-1}\right])but$$$$0<\frac{1}{2k-1}<1fork\in \left[[2,1019]\right]\Rightarrow$$$$\left[\frac{1}{2k-1}\right]=0\Rightarrow \prod _{k=1}^{1019}\left[\frac{2k}{2k-1}\right]=2$$$$$$

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