Fun with 2018 #12 (Gauss, Prince of Mathematics)
2 0 1 8 ⋅ tan ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 1 + 3 + 5 + 7 + 9 + ⋱ 1 6 9 4 1 1 ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞
What is the value of the above expression where the angle is expressed in radians?
The answer is 2018.00.
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If ∣ z ∣ ≤ 1 , Gauss's continued fraction for arctan(z) obtained from its MacLaurin series is: arctan (z) = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 1 + 3 + 5 + 7 + 9 + ⋱ ( 4 z ) 2 ( 3 z ) 2 ( 2 z ) 2 ( 1 z ) 2 z ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ Substituing z = 1 , we get the result because 4 π = arctan (1) = 1 + 3 + 5 + 7 + 9 + ⋱ 4 2 3 2 2 2 1 2 1