Combinatorial Summation!

Calculus Level 5

\[\large{ \sum_{n=0}^{\infty} \dfrac{2^{n+1}}{(2n+1) \ {2n \choose n} }}\]

If the summation above can be represented as \(\large{A \cdot \pi^B}\), where \(A,B\) are positive integers, then find the value of \((A+B)^3\)?

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