Comments by "EebstertheGreat" (@EebstertheGreat) on "The Prime Constant - Numberphile" video.

  1. Yeah, it should be 0.(01) = ¼ + 1⁄16 + 1⁄64 + · · · . The partial sums are ¼, 5⁄16, 21⁄64, . . . , which have the form n/(3n+1), so they approach ⅓.
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  2. That's pretty nice as a continuous bijection from the set of sequences of positive integers (ℕ⁺^ℕ) to the set of irrational numbers between 0 and 1 ((0,1) \ ℚ). It doesn't hit any rational numbers, because rational numbers have finite continued fractions. Another approach is to map the sequence (aₙ) ∈ ℕ^ℕ to the generalized continued fraction a₀ + 1/(1 + 1/(a₁ + 1/(1 + 1/(a₂ + · · · )))···). This maps the set of sequences of nonnegative integers onto all positive real numbers. And it's still bijective and continuous. It also preserves order (i.e. the lexicographic order on the sequences), because by taking the reciprocal twice between each term in the sequence, we effectively flip the order twice. Applying that function to the sequence of primes gives 2 + 1/(1 + 1/(3 + 1/(1 + 1/(5 + · · · )))···) = 2.79401907....
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