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EXAMPLE
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a(1) = ceiling(r(1)) = ceiling(1/tau) = ceiling(0.618...) = 1;
a(2) = ceiling(r(2)/(1 - r(1)/1) = 1;
a(3) = ceiling(r(3)/(1 - r(1)/1 - r(2)/2) = 3.
The first 3 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are
0.618..., 0.927..., 0.995...
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MATHEMATICA
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$MaxExtraPrecision = Infinity; z = 16;
r[k_] := N[1/(k*GoldenRatio), 1000]; f[x_, 0] = x;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = 1; Table[n[x, k], {k, 1, z}]
N[Sum[r[k]/n[x, k], {k, 1, 18}], 200]
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