How do I show that $$\sum_{k=1}^\infty \left({k \over 7}\right)\Big/k = \sum_{k=0}^\infty \left(\frac{1}{7k+1} + \frac{1}{7k+2} - \frac{1}{7k+3} + \frac{1}{7k+4} - \frac{1}{7k+5} - \frac{1}{7k+6}\right)$$ is $\pi/\sqrt{7}$ (this is the result Mathematica gives)
I think this can be done by expressing it in terms of sums that look like the power series for log(1+x) but I am not sure how.