$\large{S = 1 + \dfrac{1}{3} - \dfrac{1}{5} - \dfrac{1}{7} + \dfrac{1}{9} + \dfrac{1}{11} - \dfrac{1}{13} - \dfrac{1}{15} + \ldots}$

If $S$ can be expressed as $\dfrac{\pi^A}{\sqrt{B}}$ for positive integers $A,B$, submit the value of $A+B$ as your answer.