Isn't there any smart mathematician who can calculate the exact answer of this infinite summation?

Question

1-(1/4)+(1/7)-(1/10)+(1/13)-...

Answer 1

It reduces to evaluate the integral 1/(1+x^3) from 0 to 1.

(1/3)( ln(2) + pi*sqrt(3)/3 ) = 0.8356488482...

Answer 2

there are some summations whose exact values cannot be determined ( I dont know yet if this is one.) Some are best approximated by a computer.

You can start with the fact that:
1 -1/2 + 1/3 -1/4 + 1/5 -1/6 + 1/7 ... = ln2. d:

Edit: what is clear though, is that your aforementioned series is also conditionally convergent.

Answer 3

7/8

Answer 4

.........0.76...........