I Have More Bananas!

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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