Mathematica 8:

Sum[1/(i + 0) - 1/(i + 1) - 2/(i + 2) - 1/(i + 3) + 1/(i + 4) +

2/(i + 5), {i, 1, Infinity, 6}]

And a variant:

ListLinePlot[

N[Table[Sum[

1/(i + 0) - 1/(i + 1) - 2/(i + 2) - 1/(i + 3) + 1/(i + 4) +

2/(i + 5), {i, 1, Infinity, j}], {j, 1, 24}]]]

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