OFFSET
1,2
COMMENTS
Conjecture: Pi = lim_{k -> infinity} 2^{k + 1}*(1 - c_k), where the variable c_k is defined by a set of the Viète-like recurrence relations {a_1 = sqrt(2), a_k = sqrt(2 + a_{k - 1}), b_k = sqrt(2 - a_k)/a_{k + 1}, c_1 = b_1, c_k = (c_{k - 1} + b_k)/(1 - c_{k - 1}*b_k)}. From this conjecture it follows that Sum_{k >= 1} (1 - c_k) is convergent [Abrarov and Quine].
LINKS
Sanjar Abrarov, Table of n, a(n) for n = 1..104
S. M. Abrarov and B. M. Quine, A set of the Viète-like recurrence relations for the unity constant, arXiv:1702.00901 [math.GM], 2017.
FORMULA
Sum_{k >= 1} (1 - c_k) = 1.287194... , where c_k is computed by the recurrence equations a_1 = sqrt(2), a_k = sqrt(2 + a_{k - 1}), b_k = sqrt(2 - a_k)/a_{k + 1}, c_1 = b_1 and c_k = (c_{k - 1} + b_k)/(1 - c_{k - 1}*b_k).
EXAMPLE
1.287194036067924017020927807581...
MATHEMATICA
Clear[a, b, c]
a[k_] := N[Nest[Sqrt[2 + #1] &, 0, k], 100]
b[k_] := b[k] = Sqrt[2 - a[k]]/a[k + 1]
c[1] := b[1] = b[1]
c[k_] := c[k] = (c[k - 1] + b[k])/(1 - c[k - 1]*b[k])
k := 90
Print["Index k = ", k]
m := 1
Print["Power m = ", m]
(* The equation (12) *)
apprPi := 2^(k + 1)*(1 - c[k]^m)
Print["Actual value of Pi is ", N[Pi, 30]]
Print["At k = ", k, " the approximated value of Pi is ", N[apprPi, 30]]
K := 300
Print["Truncating integer K = ", K]
Print["Computing the digits ..."]
RealDigits[N[Sum[1 - c[k]^m, {k, 1, K}], 30]][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Sanjar Abrarov, Feb 06 2017
STATUS
approved