Dirichlet series converging to zero or logarithm of one

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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