Define \(f(x)\) as \[f(x) = \sum_{n=0}^{\infty} \frac{a_{n}x^{n+1}}{n+1}\] with \(a_{n} = L_{n+1}\) which \(L_{n}\) is Lucas sequence, \(L_{0}=2, L_{1}=1\) If \(f(\frac{1}{2})\) can be written as \(\ln(k)\), calculate \(k\)

Note: \(a_{n+2} = a_{n+1} + a_{n}\) and \(L_{n+2} = L_{n+1} + L_{n}\)

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