Inspired by Rohit Ner

Calculus Level 5

\[\displaystyle\lim_{N \rightarrow \infty} \sum_{n=1}^{N} \left( (-1)^{n+1} \left( 2 - \sum_{k=0}^{2n} \dfrac{(-1)^{k}}{\dbinom{2n}{k}} \right) \right)\]

If the above expression equals \(a - \ln(b),\) where \(a\) and \(b\) are positive integers, then find \(a^{3} + b^{3}.\)

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