How to calculate the infinite sum of the following series, related to binomial expansion for rational number, $r$:

$$1-\frac{r}{1!}\cdot\frac{1}{3}+\frac{r(r-1)}{2!}\cdot\frac{1}{5}-\frac{r(r-1)(r-2)}{3!}\cdot\frac{1}{7}+\ \dots$$.

I know the limit:

$$1-1/3+1/5-1/7+\ \dots = \pi/4$$

and I can calculate:

$$1+\frac{r}{1!}+\frac{r(r-1)}{2!}+\frac{r(r-1)(r-2)}{3!}+\ \dots$$

but I'm ot sure if that helps finding the solution for a starting problem..?

Thanx, Dragisa