A Weird Infinity Fractional Sequence ~ Inspired by Sourasish K

\[\begin{eqnarray} \dfrac{1}{2} + \dfrac{3}{2 \cdot 4} + \dfrac{3 \cdot 5 }{2 \cdot 4 \cdot 8 } + \dfrac{3 \cdot 5 \cdot 7 }{ 2 \cdot 4 \cdot 8 \cdot 12 } + \cdots = \text{?} \end{eqnarray} \]

Clarification: The \(n^\text{th}\) term of this series can be expressed as \( \dfrac{n}{2^{3n - 1}} \dbinom{2n}{n} \).


Inspiration.

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