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I don't think you're missing anything: Sage is not computing the infinite sum correctly. The summand is half of 1/((n+1)*(2n-1)), and that's easier to type:

sage: sum(1/((n+1)*(2*n-1)), n, 0, 1000).n()
-0.205068171626375
sage: sum(1/((n+1)*(2*n-1)), n, 0, 10000).n()
-0.204618542543703
sage: sum(1/((n+1)*(2*n-1)), n, 0, 100000).n()  # seems to be converging
-0.204573546255870
sage: sum(1/((n+1)*(2*n-1)), n, 0, oo).n()      # but not to this number
-1.09345743518226

sage: sum(1/((n+1)*(2*n-1)), n, 0, oo)
2/3*log(2) - 14/9
sage: sum(1/((n+1)*(2*n-1)), n, 0, oo).n() - sum(1/((n+1)*(2*n-1)), n, 0, 10000).n()
-0.888838892638556

The infinite sum differs from the partial sum by about 8/9, which is consistent with what Mathematica says.