400 Followers Problem - Reciprocal Summations 3

Calculus Level 5

\[\large{S = 1 + \dfrac{1}{5}- \dfrac{1}{7} - \dfrac{1}{11} + \dfrac{1}{13} + \dfrac{1}{17} - \dfrac{1}{19} - \dfrac{1}{23} + \ldots }\]

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


Also try these:
400 Followers Problem - Reciprocal Summations #1
400 Followers Problem - Reciprocal Summations #2
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