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Number of decimal digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value.
3

%I #14 Aug 11 2018 20:56:20

%S 0,-1,1,0,2,4,5,4,5,6,6,6,7,8,9,10,11,10,12,13,13,13,14,15,15,16,17,

%T 18,20,22,22,23,23,24,25,26,27,27,28,29,31,32,33,34,35,36,38,40,39,41,

%U 39,43,44,45,46,48,48,49,51,52,52,54,54,55,55,56,57,57,58

%N Number of decimal digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value.

%C Decimal expansion of log(2) in A002162.

%C For the number of correct binary digits see A317557.

%C For the similar case of number of correct decimal digits of Pi see A084407.

%H A.H.M. Smeets, <a href="/A317558/b317558.txt">Table of n, a(n) for n = 1..20000</a>

%F Lim_{n -> oo} a(n)/n = 2*log(A086702)/log(10) = 2*A100199/log(10) = 2*A240995.

%e n convergent decimal expansion a(n)

%e == ============ ==================== ====

%e 1 0 / 1 0.0 0

%e 2 1 / 1 1.0 -1

%e 3 2 / 3 0.66... 1

%e 4 7 / 10 0.7... 0

%e 5 9 / 13 0.692... 2

%e 6 61 / 88 0.69318... 4

%e 7 192 / 277 0.693140... 5

%e 8 253 / 365 0.69315... 4

%e 9 445 / 642 0.693146... 5

%e 10 1143 / 1649 0.6931473... 6

%e oo lim = log(2) 0.693147180559945... --

%t a[n_] := Block[{k = 1, a = RealDigits[Log@2, 10, n + 10][[1]], b = RealDigits[ FromContinuedFraction@ ContinuedFraction[ Log@2, n], 10, n + 10][[1]]}, While[a[[k]] == b[[k]], k++]; k - 1]; a[1] = 0; a[2] = -1; Array[a, 69] (* _Robert G. Wilson v_, Aug 09 2018 *)

%Y Cf. A002162, A016730, A086702, A100199, A240995, A317557.

%K sign,base

%O 1,5

%A _A.H.M. Smeets_, Jul 31 2018

%E a(61) onward from _Robert G. Wilson v_, Aug 09 2018