Inspired By Shubhendra Singh

\[\large \dfrac{1}{2^3 \times3!}-\dfrac{1\times 3}{2^4\times 4!}+\dfrac{1\times3\times5}{2^5\times5!} - \ldots \]

If the alternating series above equals to \( \dfrac A{24} - \dfrac B{B+1} \sqrt2 \) for positive integers \(A\) and \(B\), find the value of \(A+B\).

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