## 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;