Here is (what I think is) my program for computing $\log(2)$ with Van Wijngaarden:
nn = 100;
S = N[Table[Sum[(-1)^(n + 1)/n, {n, 1, k}], {k, 1, nn}], 20];
S = Table[
Last[S =
Table[(S[[k]] + S[[k + 1]])/2, {k, 1, Length[S] - 1}]], {i, 1,
nn - 1}];
S = Table[
Last[S =
Table[(S[[k]] + S[[k + 1]])/2, {k, 1, Length[S] - 1}]], {i, 1,
nn - 2}];
S = Table[
Last[S =
Table[(S[[k]] + S[[k + 1]])/2, {k, 1, Length[S] - 1}]], {i, 1,
nn - 3}];
S = Table[
Last[S =
Table[(S[[k]] + S[[k + 1]])/2, {k, 1, Length[S] - 1}]], {i, 1,
nn - 4}];
S = Table[
Last[S =
Table[(S[[k]] + S[[k + 1]])/2, {k, 1, Length[S] - 1}]], {i, 1,
nn - 5}];
S = Table[
Last[S =
Table[(S[[k]] + S[[k + 1]])/2, {k, 1, Length[S] - 1}]], {i, 1,
nn - 6}];
S = Table[
Last[S =
Table[(S[[k]] + S[[k + 1]])/2, {k, 1, Length[S] - 1}]], {i, 1,
nn - 7}];
S = Table[
Last[S =
Table[(S[[k]] + S[[k + 1]])/2, {k, 1, Length[S] - 1}]], {i, 1,
nn - 8}];
S = Table[
Last[S =
Table[(S[[k]] + S[[k + 1]])/2, {k, 1, Length[S] - 1}]], {i, 1,
nn - 9}];
S = Table[
Last[S =
Table[(S[[k]] + S[[k + 1]])/2, {k, 1, Length[S] - 1}]], {i, 1,
nn - 10}];
Last[S]
N[Log[2], 20]
I leave it like this so that you can make the exercise and write it in compact form
as a loop or something.
EulerSum[]
? $\endgroup$