Dirichlet series converging to zero

Mathematica


Clear[j, a1, cc, OnePlusB, n, dd, a]
OnePlusB = (1 + N[Sum[(-1)^j*(3*j)^(-1/2), {j, 1, Infinity}], 120])
a1 = N[Sum[
1/Sqrt[i] - 1/Sqrt[1 + i] - 2/Sqrt[2 + i] - 1/Sqrt[3 + i] + 1/Sqrt[
4 + i] + 2/Sqrt[5 + i], {i, 1, \[Infinity], 6}], 500]
Monitor[cc = Table[a1*OnePlusB^n, {n, 0, 1000000}];, n]
dd = 2 + Total[cc]
a1 = N[Sum[
1/Sqrt[i] - 1/Sqrt[1 + i] - dd/Sqrt[2 + i] - 1/Sqrt[3 + i] + 1/
Sqrt[4 + i] + dd/Sqrt[5 + i], {i, 1, \[Infinity], 6}], 500]

Dirichlet series converging to zero

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