A perfectly natural question ....

Calculus Level 5

It is the case that

\(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{2^{n} * (n - \frac{1}{2})} = \sqrt{a}*\ln(b + \sqrt{c})\),

where \(a,b,c\) are positive integers with \(a\) and \(c\) being square-free.

Find \(a^{b+c}\).

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