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Calculus Level 5

Let R = 2 2 2 ⋱ 6 ⋅ 7 4 ⋅ 5 2 ⋅ 3 . \Large R = \sqrt[2 \cdot 3]{\frac{2}{\sqrt[4 \cdot 5]{\frac{2}{\sqrt[6 \cdot 7]{\frac{2}{\ddots}}}}}} .

Find ⌊ 1000 R ⌋ . \left \lfloor{1000 R}\right \rfloor .


The answer is 1116.

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Max Filippov by Max Filippov , 26, Russia

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1 solution

Otto Bretscher
Feb 4, 2016

2 1 3 ! − 1 5 ! + 1 7 ! − . . . = 2 1 − sin ⁡ ( 1 ) ≈ 1.116 2^{\frac{1}{3!}-\frac{1}{5!}+\frac{1}{7!}-...}=2^{1-\sin(1)}\approx 1.116 . The answer is 1116 \boxed{1116}

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