text: If I understand this right, you have a(n) = A^(-2n + 1)/(2n - 1), so\n\nS(n) = 1/A + 1/(3A^3) + 1/(5A^5) + ...\n\nLet f(x) =x^(-2n + 1)/(2n - 1) = x^(-1) + x^(-3)/3 + x^(-5)/5 + ...\n\nThen f'(x) (derivative of x) is\n\nf'(x) = -x^(-2) - x^(-4) - x^(-6) - ...\n\nThis is a power series and can be summed over the range it converges as -x^(-2)/(1 - x^-2), then you need to integrate to get the actual function.