[VA] SRC #007 - 2020 April 1st Ramblings

[Albert Chan]

(04-01-2020 06:52 PM)Valentin Albillo Wrote:  6) The following expression (where N > 0 is an integer and log2 is the natural logarithm of 2 = 0.693+):

      Ceil(2/(2^1/N - 1)) = [2*N/log2]

seems to be an identity for all integer values of n ...

Let gap = f(N) = 2*N/log2 - 2/(2^1/N - 1)
Let w = log(2)/N > 0     → f(N) = g(w) = 2/w - 2/(e^w-1)

Since e^w-1 = w + w²/2! + w³/3! + ... > w, we have f(N) = g(w) > 0

f(1) ≈ 0.8854
f(2) ≈ 0.9424
f(3) ≈ 0.9615
...
f(2020) ≈ 0.9999

Apply L'Hospital's rule for limit(g(w), w=0):

2*(e^w-1 - w) / (w * (e^w-1))
⇒ 2*(e^w-1) / (w*e^w + (e^w-1))
⇒ 2*e^w / (w*e^w + e^w + e^w)
= 2 / (w + 2)

→ f(∞) = g(0) = 2 / (0+2) = 1

→ Ceil(2/(2^1/N - 1)) ≥ [2*N/log2]     // LHS > RHS if {2*N/log2} > f(N)

Example: 37th convergents of log2/2 = 777451915729368 / 2243252046704767

With N = 777451915729368     // note: this may not be the first exception case

LHS = Ceil ( 2243252046704766.000000000000000106 ... ) = 2243252046704767
RHS = Floor(2243252046704766.999999999999999957 ... ) = 2243252046704766