Natural numbers, harmonic numbers, logarithms and partial sums of square roots with BBP type formulas

With this Mathematica 8 program:


(*program start*)
Clear[s, i, n, j]
s = 1;
i = 1;
j = 0;
Sum[1/(n + 0)^s - 1/(n + 1)^s, {n, 1, Infinity, i + j}]
Sum[1/(n + 0)^s + 1/(n + 1)^s - 2/(n + 2)^s, {n, 1, Infinity, i + 2*j}]
Sum[1/(n + 0)^s + 1/(n + 1)^s + 1/(n + 2)^s - 3/(n + 3)^s, {n, 1,
Infinity, i + 3*j}]
Sum[1/(n + 0)^s + 1/(n + 1)^s + 1/(n + 2)^s + 1/(n + 3)^s -
4/(n + 4)^s, {n, 1, Infinity, i + 4*j}]
Sum[1/(n + 0)^s + 1/(n + 1)^s + 1/(n + 2)^s + 1/(n + 3)^s +
1/(n + 4)^s - 5/(n + 5)^s, {n, 1, Infinity, i + 5*j}]
Sum[1/(n + 0)^s + 1/(n + 1)^s + 1/(n + 2)^s + 1/(n + 3)^s +
1/(n + 4)^s + 1/(n + 5)^s - 6/(n + 6)^s, {n, 1, Infinity, i + 6*j}]
Sum[1/(n + 0)^s + 1/(n + 1)^s + 1/(n + 2)^s + 1/(n + 3)^s +
1/(n + 4)^s + 1/(n + 5)^s + 1/(n + 6)^s - 7/(n + 7)^s, {n, 1,
Infinity, i + 7*j}]
(*program end*)

I noticed some probably known facts:

With parameters:
s = 1;
i = 1;
j = 0;
-> 1,2,3,4,5,6,7…

With parameters:
s = 2;
i = 1;
j = 0;
-> 1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140 …
which starts as the harmonic numbers.

With parameters:
s = 1;
i = 1;
j = 1;
-> Log[2], Log[3], Log[4], Log[5], Log[6], Log[7], Log[8] …
(Jaume Oliver Lafont has commented on this)

With parameters:
s = 1/2;
i = 1;
j = 0;
and after wrapping the answer with N[answer,20] I get these
inaccurate numbers accompanied with error messages for every number:

2.4142135623730931752 = Sqrt[1] + Sqrt[2],
4.1462643699419685951 = Sqrt[1] + Sqrt[2] + Sqrt[3],
6.1462643699419710784 = Sqrt[1] + Sqrt[2] + Sqrt[3] + Sqrt[4],
8.3823323474417526706 = Sqrt[1] + Sqrt[2] + Sqrt[3] + Sqrt[4] + Sqrt[5],
10.831822090224927022 = Sqrt[1] + Sqrt[2] + Sqrt[3] + Sqrt[4] + Sqrt[5] + Sqrt[6],
13.477573401289513240 = Sqrt[1] + Sqrt[2] + Sqrt[3] + Sqrt[4] + Sqrt[5] + Sqrt[6] + Sqrt[7], …

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