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EXAMPLE
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The algorithm at A226049, with r = sqrt(2), f(n) = 1/(2n+1), gives
1/3 + 1/5 + 1/7 + ... + 1/(1+2*17) - 1/(1+2*38) + 1/(1+2*2558) - ...
which converges to sqrt(2). The 17th partial sum differs from sqrt(2) by less than 10^(-500).
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MATHEMATICA
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$MaxExtraPrecision = Infinity; z = 9; f[n_] := 1/(2 n + 1); g[n_] := (1/n - 1)/2; r =
Sqrt[2]; s = 0; a[1] = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; p = Sum[f[n], {n, 1, a[1]}]; a[2] = Floor[g[p - r]]; a[n_] := Floor[g[((-1)^n) (p - r - Sum[((-1)^k) f[a[k]], {k, 2, n - 1}])]]; Table[a[k], {k, 1, z}]
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