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

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

Answered by Bird last updated on 26/Oct/20

\prod_{k = 1}^{1019} [\frac{2 k}{2 k - 1}] = \prod_{k = 1}^{1019} [\frac{2 k - 1 + 1}{2 k - 1}]

= \prod_{k = 1}^{1019} [1 + \frac{1}{2 k - 1}]

= \prod_{k = 1}^{1019} (1 + [\frac{1}{2 k - 1}])

= 2 \prod_{k = 2}^{1019} (1 + [\frac{1}{2 k - 1}]) b u t

0 < \frac{1}{2 k - 1} < 1 f o r k \in [[2, 1019]] \Rightarrow

[\frac{1}{2 k - 1}] = 0 \Rightarrow \prod_{k = 1}^{1019} [\frac{2 k}{2 k - 1}] = 2

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